Brief Contents
PART ONE
Introduction and Review
Chapter 1
Chapter 2
Chapter 3
Economic Questions and Data
Review of Probability 13
Review of Statistics 61
PART TWO
Fundamentals of Regression Analysis
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Linear Regression with One Regressor 101
Regression with a Single Regressor: Hypothesis Tests
and Confidence Intervals 136
Linear Regression with Multiple Regressors 169
Hypothesis Tests and Confidence Intervals in Multiple Regression
Nonlinear Regression Functions 235
Assessing Studies Based on Multiple Regression 288
PART THREE
Further Topics in Regression Analysis
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Regression with Panel Data 319
Regression with a Binary Dependent Variable 350
Instrumental Variables Regression 385
Experiments and Quasi-Experiments 432
Prediction with Many Regressors and Big Data 472
PART FOUR
Regression Analysis of Economic Time Series Data
Chapter 15
Chapter 16
Chapter 17
Introduction to Time Series Regression and Forecasting
Estimation of Dynamic Causal Effects 567
Additional Topics in Time Series Regression 607
PART FIVE
Regression Analysis of Economic Time Series Data
Chapter 18
Chapter 19
The Theory of Linear Regression with One Regressor
The Theory of Multiple Regression 671
1
205
512
645
v
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Contents
PART THREE
Further Topics in Regression Analysis
CHAPTER 10
Regression with Panel Data
10.1
Panel Data
319
320
Example: Traffic Deaths and Alcohol Taxes
320
10.2
Panel Data with Two Time Periods: “Before and After” Comparisons
10.3
Fixed Effects Regression
325
The Fixed Effects Regression Model
Estimation and Inference 327
Application to Traffic Deaths 328
10.4
325
Regression with Time Fixed Effects
Time Effects Only 329
Both Entity and Time Fixed Effects
10.5
323
329
330
The Fixed Effects Regression Assumptions and Standard Errors for Fixed
Effects Regression 332
The Fixed Effects Regression Assumptions 332
Standard Errors for Fixed Effects Regression 334
10.6
Drunk Driving Laws and Traffic Deaths
10.7
Conclusion
335
339
APPENDIX 10.1!The State Traffic Fatality Data Set
345
APPENDIX 10.2!Standard Errors for Fixed Effects Regression
CHAPTER 11
11.1
Regression with a Binary Dependent Variable
346
350
Binary Dependent Variables and the Linear Probability Model
351
Binary Dependent Variables 351
The Linear Probability Model 353
11.2
Probit and Logit Regression
355
Probit Regression 355
Logit Regression 359
Comparing the Linear Probability, Probit, and Logit Models
11.3
361
Estimation and Inference in the Logit and Probit Models
362
Nonlinear Least Squares Estimation 362
Maximum Likelihood Estimation 363
Measures of Fit 364
11.4
Application to the Boston HMDA Data
11.5
Conclusion
365
371
APPENDIX 11.1!The Boston HMDA Data Set
379
APPENDIX 11.2!Maximum Likelihood Estimation 379
APPENDIX 11.3!Other Limited Dependent Variable Models
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Contents
CHAPTER 12
12.1
Instrumental Variables Regression
The IV Estimator with a Single Regressor and a Single Instrument
The IV Model and Assumptions 386
The Two Stage Least Squares Estimator 387
Why Does IV Regression Work? 387
The Sampling Distribution of the TSLS Estimator
Application to the Demand for Cigarettes 393
12.2
385
The General IV Regression Model
386
392
395
TSLS in the General IV Model 397
Instrument Relevance and Exogeneity in the General IV Model 398
The IV Regression Assumptions and Sampling Distribution of the TSLS Estimator
Inference Using the TSLS Estimator 400
Application to the Demand for Cigarettes 401
12.3
Checking Instrument Validity
399
402
Assumption 1: Instrument Relevance 402
Assumption 2: Instrument Exogeneity 404
12.4
Application to the Demand for Cigarettes
408
12.5
Where Do Valid Instruments Come From?
412
12.6
Three Examples
413
Conclusion
417
APPENDIX 12.1!The Cigarette Consumption Panel Data Set
424
APPENDIX 12.2!Derivation of the Formula for the TSLS Estimator
in Equation (12.4) 424
APPENDIX 12.3!Large-Sample Distribution of the TSLS Estimator 425
APPENDIX 12.4!Large-Sample Distribution of the TSLS Estimator
When the Instrument Is Not Valid 426
APPENDIX 12.5!Instrumental Variables Analysis with Weak Instruments
APPENDIX 12.6 !TSLS with Control Variables 429
CHAPTER 13
13.1
Experiments and Quasi-Experiments
427
432
Potential Outcomes, Causal Effects, and Idealized Experiments
433
Potential Outcomes and the Average Causal Effect 433
Econometric Methods for Analyzing Experimental Data 434
13.2
Threats to Validity of Experiments
436
Threats to Internal Validity 436
Threats to External Validity 439
13.3
Experimental Estimates of the Effect of Class Size Reductions
440
Experimental Design 440
Analysis of the STAR Data 441
Comparison of the Observational and Experimental Estimates of Class Size Effects
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CHAPTER
10
Regression with Panel Data
M
ultiple regression is a powerful tool for controlling for the effect of variables on
which we have data. If data are not available for some of the variables, however,
they cannot be included in the regression, and the OLS estimators of the regression
coefficients could have omitted variable bias.
This chapter describes a method for controlling for some types of omitted
variables without actually observing them. This method requires a specific type of
data, called panel data, in which each observational unit, or entity, is observed at
two!or more time periods. By studying changes in the dependent variable over
time, it is possible to eliminate the effect of omitted variables that differ across
entities but are constant over time.
The empirical application in this chapter concerns drunk driving: What are the
effects of alcohol taxes and drunk driving laws on traffic fatalities? We address this
question using data on traffic fatalities, alcohol taxes, drunk driving laws, and
related variables for the 48 contiguous U.S. states for each of the seven years from
1982 to 1988. This panel data set lets us control for unobserved variables that differ
from one state to the next, such as prevailing cultural attitudes toward drinking and
driving, but do not change over time. It also allows us to control for variables that
vary through time, like improvements in the safety of new cars, but do not vary
across!states.
Section 10.1 describes the structure of panel data and introduces the drunk
driving data set. Fixed effects regression, the main tool for regression analysis of
panel data, is an extension of multiple regression that exploits panel data to
control!for variables that differ across entities but are constant over time. Fixed
effects regression is introduced in Sections 10.2 and 10.3, first for the case of only
two time periods and then for multiple time periods. In Section 10.4, these
methods are extended to incorporate so-called time fixed effects, which control
for!unobserved variables that are constant across entities but change over time.
Section 10.5 discusses the panel data regression assumptions and standard errors
for!panel data regression. In Section 10.6, we use these methods to study the
effect!of alcohol taxes and drunk driving laws on traffic deaths.
319
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CHAPTER 10!!Regression with Panel Data
KEY CONCEPT
10.1
Notation for Panel Data
Panel data consist of observations on the same n entities at two or more time
periods T, as is illustrated in Table 1.3. If the data set contains observations on the
variables X and Y, then the data are denoted
1Xit, Yit 2, i = 1, c, n, and t = 1, c, T,
(10.1)
where the first subscript, i, refers to the entity being observed and the second
subscript, t, refers to the date at which it is observed.
10.1 Panel Data
Recall from Section 1.3 that panel data (also called longitudinal data) refers to data
for n different entities observed at T different time periods. The state traffic fatality
data studied in this chapter are panel data. Those data are for n = 48 entities (states),
where each entity is observed in T = 7 time periods (each of the years 1982, . . . ,
1988), for a total of 7 * 48 = 336 observations.
When describing cross-sectional data, it was useful to use a subscript to denote
the entity; for example, Yi referred to the variable Y for the i th entity. When describing
panel data, we need some additional notation to keep track of both the entity and
the time period. We do so by using two subscripts rather than one: The first, i, refers
to the entity, and the second, t, refers to the time period of the observation. Thus Yit
denotes the variable Y observed for the i th of n entities in the t th of T periods. This
notation is summarized in Key Concept 10.1.
Some additional terminology associated with panel data describes whether some
observations are missing. A balanced panel has all its observations; that is, the variables are observed for each entity and each time period. A panel that has some missing data for at least one time period for at least one entity is called an unbalanced
panel. The traffic fatality data set has data for all 48 contiguous U.S. states for all
seven years, so it is balanced. If, however, some data were missing (for example, if we
did not have data on fatalities for some states in 1983), then the data set would be
unbalanced. The methods presented in this chapter are described for a balanced
panel; however, all these methods can be used with an unbalanced panel, although
precisely how to do so in practice depends on the regression software being used.
Example: Traffic Deaths and Alcohol Taxes
There are approximately 40,000 highway traffic fatalities each year in the United
States. Approximately one-fourth of fatal crashes involve a driver who was drinking,
and this fraction rises during peak drinking periods. One study (Levitt and Porter,
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10.1!!Panel Data
321
2001) estimates that as many as 25% of drivers on the road between 1 a.m. and 3 a.m.
have been drinking and that a driver who is legally drunk is at least 13 times as likely
to cause a fatal crash as a driver who has not been drinking.
In this chapter, we study how effective various government policies designed to
discourage drunk driving actually are in reducing traffic deaths. The panel data set
contains variables related to traffic fatalities and alcohol, including the number of
traffic fatalities in each state in each year, the type of drunk driving laws in each state
in each year, and the tax on beer in each state. The measure of traffic deaths we use
is the fatality rate, which is the number of annual traffic deaths per 10,000 people in
the population in the state. The measure of alcohol taxes we use is the “real” tax on
a case of beer, which is the beer tax, put into 1988 dollars by adjusting for inflation.1
The data are described in more detail in Appendix 10.1.
Figure 10.1a is a scatterplot of the data for 1982 on two of these variables, the
fatality rate and the real tax on a case of beer. A point in this scatterplot represents
the fatality rate in 1982 and the real beer tax in 1982 for a given state. The OLS
regression line obtained by regressing the fatality rate on the real beer tax is also
plotted in the figure; the estimated regression line is
FatalityRate = 2.01 + 0.15 BeerTax
10.152 10.132
11982 data2.
FatalityRate = 1.86 + 0.44 BeerTax
10.112 10.132
11988 data2.
(10.2)
The coefficient on the real beer tax is positive but not statistically significant at the
10% level.
Because we have data for more than one year, we can reexamine this relationship for another year. This is done in Figure 10.1b, which is the same scatterplot as
before except that it uses the data for 1988. The OLS regression line through these
data is
(10.3)
In contrast to the regression using the 1982 data, the coefficient on the real beer
tax is statistically significant at the 1% level (the t-statistic is 3.43). Curiously, the
estimated coefficients for the 1982 and the 1988 data are positive: Taken literally,
higher real beer taxes are associated with more, not fewer, traffic fatalities.
Should we conclude that an increase in the tax on beer leads to more traffic
deaths? Not necessarily, because these regressions could have substantial omitted
variable bias. Many factors affect the fatality rate, including the quality of the
automobiles driven in the state, whether the state highways are in good repair,
whether most driving is rural or urban, the density of cars on the road, and whether
it is socially acceptable to drink and drive. Any of these factors may be correlated
1
To make the taxes comparable over time, they are put into 1988 dollars using the Consumer Price Index
(CPI). For example, because of inflation, a tax of $1 in 1982 corresponds to a tax of $1.23 in 1988 dollars.
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CHAPTER 10!!Regression with Panel Data
FIGURE 10.1
The Traffic Fatality Rate and the Tax on Beer
Figure 10.1a is a scatterplot of
traffic fatality rates and the real
tax on a case of beer (in 1988
dollars) for 48 states in 1982.
Figure 10.1b shows the data
for 1988. Both plots show a
positive relationship between
the fatality rate and the real
beer tax.
Fatality rate
(fatalities per 10,000)
4.5
4.0
3.5
3.0
FatalityRate = 2.01 + 0.15 BeerTax
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0.5
1.0
1.5
(a) 1982 data
2.0
2.5
3.0
Beer tax
(dollars per case $1988)
Fatality rate
(fatalities per 10,000)
4.5
4.0
3.5
FatalityRate = 1.86 + 0.44 BeerTax
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Beer tax
(dollars per case $1988)
(b) 1988 data
with alcohol taxes, and if so, this will lead to omitted variable bias. One approach
to these potential sources of omitted variable bias would be to collect data on all
these variables and add them to the annual cross-sectional regressions in Equations (10.2) and (10.3). Unfortunately, some of these variables, such as the cultural
acceptance of drinking and driving, might be very hard or even impossible to
measure.
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10.2!!Panel Data with Two Time Periods: “Before and After” Comparisons
323
If these factors remain constant over time in a given state, however, then another
route is available. Because we have panel data, we can, in effect, hold these factors
constant even though we cannot measure them. To do so, we use OLS regression with
fixed effects.
10.2 Panel Data with Two Time Periods:
“Before and After” Comparisons
When data for each state are obtained for T = 2 time periods, it is possible to compare values of the dependent variable in the second period to values in the first
period. By focusing on changes in the dependent variable, this “before and after” or
“differences” comparison, in effect, holds constant the unobserved factors that differ
from one state to the next but do not change over time within the state.
Let Zi be a variable that determines the fatality rate in the i th state but does not
change over time (so the t subscript is omitted). For example, Zi might be the local
cultural attitude toward drinking and driving, which changes slowly and thus could
be considered to be constant between 1982 and 1988. Accordingly, the population
linear regression relating Zi and the real beer tax to the fatality rate is
FatalityRateit = b0 + b1BeerTaxit + b2Zi + u it,
(10.4)
where u it is the error term, i = 1, c, n, and t = 1, c, T.
Because Zi does not change over time, in the regression model in Equation (10.4)
it will not produce any change in the fatality rate between 1982 and 1988. Thus, in this
regression model, the influence of Zi can be eliminated by analyzing the change in
the fatality rate between the two periods. To see this mathematically, consider
Equation (10.4) for each of the two years 1982 and 1988:
FatalityRatei1982 = b0 + b1BeerTaxi1982 + b2Zi + u i1982,
(10.5)
FatalityRatei1988 = b0 + b1BeerTaxi1988 + b2Zi + u i1988.
(10.6)
Subtracting Equation (10.5) from Equation (10.6) eliminates the effect of Zi:
FatalityRatei1988 - FatalityRatei1982
= b1 1BeerTaxi1988 - BeerTaxi1982 2 + u i1988 - u i1982.
(10.7)
This specification has an intuitive interpretation. Cultural attitudes toward drinking
and driving affect the level of drunk driving and thus the traffic fatality rate in a state.
If, however, they did not change between 1982 and 1988, then they did not produce
any change in fatalities in the state. Rather, any changes in traffic fatalities over time
must have arisen from other sources. In Equation (10.7), these other sources are
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CHAPTER 10!!Regression with Panel Data
changes in the tax on beer and changes in the error term (which captures changes in
other factors that determine traffic deaths).
Specifying the regression in changes in Equation (10.7) eliminates the effect of
the unobserved variables Zi that are constant over time. In other words, analyzing
changes in Y and X has the effect of controlling for variables that are constant over
time, thereby eliminating this source of omitted variable bias.
Figure 10.2 presents a scatterplot of the change in the fatality rate between 1982
and 1988 against the change in the real beer tax between 1982 and 1988 for the 48
states in our data set. A point in Figure 10.2 represents the change in the fatality rate
and the change in the real beer tax between 1982 and 1988 for a given state. The OLS
regression line, estimated using these data and plotted in the figure, is
FatalityRate1988 - FatalityRate1982 = -0.072 - 1.041BeerTax1988 - BeerTax1982 2.
10.0652 10.362
(10.8)
Including an intercept in Equation (10.8) allows for the possibility that the mean
change in the fatality rate, in the absence of a change in the real beer tax, is nonzero.
For example, the negative intercept 1 -0.0722 could reflect improvements in auto
safety between 1982 and 1988 that reduced the average fatality rate.
In contrast to the cross-sectional regression results, the estimated effect of a change
in the real beer tax is negative, as predicted by economic theory. The hypothesis that
the population slope coefficient is 0 is rejected at the 5% significance level. According
to this estimated coefficient, an increase in the real beer tax by $1 per case reduces the
traffic fatality rate by 1.04 deaths per 10,000 people. This estimated effect is very large:
FIGURE 10.2 Changes in Fatality Rates and Beer Taxes from 1982 to 1988
This is a scatterplot of the
change in the
traffic fatality rate
and the change
in the real beer
tax between
1982 and 1988
for 48 states.
There is a negative relationship
between changes
in the fatality rate
and changes in
the beer tax.
Change in fatality rate
(fatalities per 10,000)
1.0
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FatalityRate1988 – FatalityRate1982 = –0.072 – 1.04(BeerTax1988 – BeerTax1982 )
0.5
0.0
–0.5
–1.0
–1.5
–0.6
–0.4
–0.2
0.0
0.2
0.4
0.6
Change in beer tax
(dollars per case $1988)
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10.3!!Fixed Effects Regression
325
The average fatality rate is approximately 2 in these data (that is, 2 fatalities per year
per 10,000 members of the population), so the estimate suggests that traffic fatalities
can be cut in half merely by increasing the real tax on beer by $1 per case.
By examining changes in the fatality rate over time, the regression in Equation
(10.8) controls for fixed factors such as cultural attitudes toward drinking and driving.
But there are many factors that influence traffic safety, and if they change over time
and are correlated with the real beer tax, then their omission will produce omitted
variable bias. In Section 10.6, we undertake a more careful analysis that controls for
several such factors, so for now it is best to refrain from drawing any substantive
conclusions about the effect of real beer taxes on traffic fatalities.
This “before and after” or “differences” analysis works when the data are observed
in two different years. Our data set, however, contains observations for seven different
years, and it seems foolish to discard those potentially useful additional data. But the
“before and after” method does not apply directly when T 7 2. To analyze all the
observations in our panel data set, we use the method of fixed effects regression.
10.3 Fixed Effects Regression
Fixed effects regression is a method for controlling for omitted variables in panel
data when the omitted variables vary across entities (states) but do not change over
time. Unlike the “before and after” comparisons of Section 10.2, fixed effects regression can be used when there are two or more time observations for each entity.
The fixed effects regression model has n different intercepts, one for each entity.
These intercepts can be represented by a set of binary (or indicator) variables. These
binary variables absorb the influences of all omitted variables that differ from one
entity to the next but are constant over time.
The Fixed Effects Regression Model
Consider the regression model in Equation (10.4) with the dependent variable
(FatalityRate) and observed regressor (BeerTax) denoted as Yit and Xit, respectively:
Yit = b0 + b1Xit + b2Zi + u it,
(10.9)
where Zi is an unobserved variable that varies from one state to the next but does
not change over time (for example, Zi represents cultural attitudes toward drinking
and driving). We want to estimate b1, the effect on Y of X, holding constant the unobserved state characteristics Z.
Because Zi varies from one state to the next but is constant over time, the population regression model in Equation (10.9) can be interpreted as having n intercepts,
one for each state. Specifically, let ai = b0 + b2Zi. Then Equation (10.9) becomes
Yit = b1Xit + ai + u it.
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CHAPTER 10!!Regression with Panel Data
Equation (10.10) is the fixed effects regression model, in which a1, c, an are treated
as unknown intercepts to be estimated, one for each state. The interpretation of ai as
a state-specific intercept in Equation (10.10) comes from considering the population
regression line for the i th state; this population regression line is ai + b1Xit. The slope
coefficient of the population regression line, b1, is the same for all states, but the
intercept of the population regression line varies from one state to the next.
Because the intercept ai in Equation (10.10) can be thought of as the “effect” of
being in entity i (in the current application, entities are states), the terms a1, c, an
are known as entity fixed effects. The variation in the entity fixed effects comes from
omitted variables that, like Zi in Equation (10.9), vary across entities but not over
time.
The state-specific intercepts in the fixed effects regression model also can be
expressed using binary variables to denote the individual states. Section 8.3 considered the case in which the observations belong to one of two groups and the population regression line has the same slope for both groups but different intercepts (see
Figure 8.8a). That population regression line was expressed mathematically using a
single binary variable indicating one of the groups (case 1 in Key Concept 8.4). If we
had only two states in our data set, that binary variable regression model would apply
here. Because we have more than two states, however, we need additional binary
variables to capture all the state-specific intercepts in Equation (10.10).
To develop the fixed effects regression model using binary variables, let D1i be
a binary variable that equals 1 when i = 1 and equals 0 otherwise, let D2i equal 1
when i = 2 and equal 0 otherwise, and so on. We cannot include all n binary variables
plus a common intercept, for if we do, the regressors will be perfectly multicollinear
(this is the dummy variable trap of Section 6.7), so we arbitrarily omit the binary
variable D1i for the first entity. Accordingly, the fixed effects regression model in
Equation (10.10) can be written equivalently as
Yit = b0 + b1Xit + g2D2i + g3D3i + g + gnDni + u it,
(10.11)
where b0, b1, g2, c , gn are unknown coefficients to be estimated. To derive the
relationship between the coefficients in Equation (10.11) and the intercepts in
Equation (10.10), compare the population regression lines for each state in the two
equations. In Equation (10.11), the population regression equation for the first state
is b0 + b1Xit, so a1 = b0. For the second and remaining states, it is b0 + b1Xit + gi,
so ai = b0 + gi for i Ú 2.
Thus there are two equivalent ways to write the fixed effects regression model,
Equations (10.10) and (10.11). In Equation (10.10), it is written in terms of n statespecific intercepts. In Equation (10.11), the fixed effects regression model has a common intercept and n - 1 binary regressors. In both formulations, the slope coefficient
on X is the same from one state to the next. The state-specific intercepts in Equation
(10.10) and the binary regressors in Equation (10.11) have the same source: the
unobserved variable Zi that varies across states but not over time.
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10.3!!Fixed Effects Regression
The Fixed Effects Regression Model
KEY CONCEPT
10.2
The fixed effects regression model is
Yit = b1X1,it + g + bkXk,it + ai + u it,
327
(10.12)
where i = 1, c, n; t = 1, c, T; X1,it is the value of the first regressor for entity i
in time period t, X2,it is the value of the second regressor, and so forth; and
a1, c, an are entity-specific intercepts.
Equivalently, the fixed effects regression model can be written in terms of a common intercept, the X’s, and n - 1 binary variables representing all but one entity:
Yit = b0 + b1X1, it + g + bkXk, it + g2D2i
+ g3D3i + g + gnDni + u it,
(10.13)
where D2i = 1 if i = 2 and D2i = 0 otherwise, and so forth.
Extension to multiple X’s. If there are other observed determinants of Y that are
correlated with X and that change over time, then these should also be included in
the regression to avoid omitted variable bias. Doing so results in the fixed effects
regression model with multiple regressors, summarized in Key Concept 10.2.
Estimation and Inference
In principle, the binary variable specification of the fixed effects regression model
[Equation (10.13)] can be estimated by OLS. This regression, however, has k + n
regressors (the k X’s, the n - 1 binary variables, and the intercept), so in practice this
OLS regression is tedious or, in some software packages, impossible to implement if
the number of entities is large. Econometric software therefore has special routines
for OLS estimation of fixed effects regression models. These special routines are
equivalent to using OLS on the full binary variable regression, but they are faster
because they employ some mathematical simplifications that arise in the algebra of
fixed effects regression.
The “entity-demeaned” OLS algorithm. Regression software typically computes
the!OLS fixed effects estimator in two steps. In the first step, the entity-specific
average is subtracted from each variable. In the second step, the regression is
estimated using “entity-demeaned” variables. Specifically, consider the case of a
single regressor in the version of the fixed effects model in Equation (10.10), and take
the average of both sides of Equation (10.10); then Yi = b1Xi + ai + u i, where
T
Yi = 11>T2 g t = 1Yit , and Xi and u i are defined similarly. Thus Equation (10.10)
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CHAPTER 10!!Regression with Panel Data
∼
∼
implies that Yit - Yi = b1 1Xit - Xi 2 + 1u it - u i 2. Let Yit = Yit - Yi, Xit = Xit - Xi
and ∼
u it = u it - u i; accordingly,
∼
∼
Yit = b1Xit + ∼
u it.
(10.14)
Thus b1 can be estimated by the OLS regression of the “entity-demeaned” variables
∼
∼
Yit on Xit. In fact, this estimator is identical to the OLS estimator of b1 obtained by
estimation of the fixed effects model in Equation (10.11) using n - 1 binary variables (Exercise 19.6).
The “before and after” (differences) regression versus the binary variables specification. Although Equation (10.11) with its binary variables looks quite different
from the “before and after” regression model in Equation (10.7), in the special case
that T = 2 the OLS estimator of b1 from the binary variable specification and that
from the “before and after” specification are identical if the intercept is excluded
from the “before and after” specification. Thus, when T = 2, there are three ways to
estimate b1 by OLS: the “before and after” specification in Equation (10.7) (without
an intercept), the binary variable specification in Equation (10.11), and the entitydemeaned specification in Equation (10.14). These three methods are equivalent;
that is, they produce identical OLS estimates of b1 (Exercise 10.11).
The sampling distribution, standard errors, and statistical inference. In multiple
regression with cross-sectional data, if the four least squares assumptions in Key Concept 6.4 hold, then the sampling distribution of the OLS estimator is normal in large
samples. The variance of this sampling distribution can be estimated from the data, and
the square root of this estimator of the variance—that is, the standard error—can be
used to test hypotheses using a t-statistic and to construct confidence intervals.
Similarly, in multiple regression with panel data, if a set of assumptions—called
the fixed effects regression assumptions—holds, then the sampling distribution of the
fixed effects OLS estimator is normal in large samples, the variance of that distribution can be estimated from the data, the square root of that estimator is the standard
error, and the standard error can be used to construct t-statistics and confidence
intervals. Given the standard error, statistical inference—testing hypotheses (including joint hypotheses using F-statistics) and constructing confidence intervals—proceeds in exactly the same way as in multiple regression with cross-sectional data.
The fixed effects regression assumptions and standard errors for fixed effects
regression are discussed further in Section 10.5.
Application to Traffic Deaths
The OLS estimate of the fixed effects regression line relating the real beer tax to the
fatality rate, based on all 7 years of data (336 observations), is
FatalityRate = -0.66 BeerTax + state fixed effects,
10.292
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10.4!!Regression with Time Fixed Effects
329
where, as is conventional, the estimated state fixed intercepts are not listed to save
space and because they are not of primary interest in this application.
Like the “before and after” specification in Equation (10.8), the estimated coefficient in the fixed effects regression in Equation (10.15) is negative, so, as predicted
by economic theory, higher real beer taxes are associated with fewer traffic deaths,
which is the opposite of what we found in the initial cross-sectional regressions of
Equations (10.2) and (10.3). The two regressions are not identical because the “before
and after” regression in Equation (10.8) uses only the data for 1982 and 1988 (specifically, the difference between those two years), whereas the fixed effects regression in
Equation (10.15) uses the data for all 7 years. Because of the additional observations,
the standard error is smaller in Equation (10.15) than in Equation (10.8).
Including state fixed effects in the fatality rate regression lets us avoid omitted
variables bias arising from omitted factors, such as cultural attitudes toward drinking
and driving, that vary across states but are constant over time within a state. Still, a
skeptic might suspect that other factors could lead to omitted variables bias. For
example, over this period cars were getting safer, and occupants were increasingly
wearing seat belts; if the real tax on beer rose, on average, during the mid-1980s, then
BeerTax could be picking up the effect of overall automobile safety improvements.
If, however, safety improvements evolved over time but were the same for all states,
then we can eliminate their influence by including time fixed effects.
10.4 Regression with Time Fixed Effects
Just as fixed effects for each entity can control for variables that are constant over
time but differ across entities, so time fixed effects can control for variables that are
constant across entities but evolve over time.
Because safety improvements in new cars are introduced nationally, they serve
to reduce traffic fatalities in all states. So it is plausible to think of automobile safety
as an omitted variable that changes over time but has the same value for all states.
The population regression in Equation (10.9) can be modified to make explicit the
effect of automobile safety, which we will denote St:
Yit = b0 + b1Xit + b2Zi + b3St + u it,
(10.16)
where St is unobserved and where the single t subscript emphasizes that safety
changes over time but is constant across states. Because b3St represents variables that
determine Yit, if St is correlated with Xit, then omitting St from the regression leads to
omitted variable bias.
Time Effects Only
For the moment, suppose that the variables Zi are not present, so that the term b2Zi
can be dropped from Equation (10.16), although the term b3St remains. Our objective
is to estimate b1, controlling for St.
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CHAPTER 10!!Regression with Panel Data
Although St is unobserved, its influence can be eliminated because it varies over time
but not across states, just as it is possible to eliminate the effect of Zi, which varies across
states but not over time. In the entity fixed effects model, the presence of Zi leads to the
fixed effects regression model in Equation (10.10), in which each state has its own intercept (or fixed effect). Similarly, because St varies over time but not over states, the presence of St leads to a regression model in which each time period has its own intercept.
The time fixed effects regression model with a single X regressor is
Yit = b1Xit + lt + u it.
(10.17)
This model has a different intercept, lt, for each time period. The intercept lt in
Equation (10.17) can be thought of as the “effect” on Y of year t (or, more generally,
time period t), so the terms l1, c, lT are known as time fixed effects. The variation
in the time fixed effects comes from omitted variables that, like St in Equation (10.16),
vary over time but not across entities.
Just as the entity fixed effects regression model can be represented using n - 1
binary indicators, so, too, can the time fixed effects regression model be represented
using T - 1 binary indicators:
Yit = b0 + b1Xit + d2B2t + g + dTBTt + u it,
(10.18)
where d2, c, dT are unknown coefficients and where B2t = 1 if t = 2 and B2t = 0
otherwise, and so forth. As in the fixed effects regression model in Equation (10.11),
in this version of the time effects model the intercept is included, and the first binary
variable 1B1t 2 is omitted to prevent perfect multicollinearity.
When there are additional observed “X” regressors, then these regressors appear
in Equations (10.17) and (10.18) as well.
In the traffic fatalities regression, the time fixed effects specification allows us to
eliminate bias arising from omitted variables like nationally introduced safety standards that change over time but are the same across states in a given year.
Both Entity and Time Fixed Effects
If some omitted variables are constant over time but vary across states (such as cultural
norms), while others are constant across states but vary over time (such as national
safety standards), then it is appropriate to include both entity (state) and time effects.
The combined entity and time fixed effects regression model is
Yit = b1Xit + ai + lt + u it,
(10.19)
where ai is the entity fixed effect and lt is the time fixed effect. This model can
equivalently be represented using n - 1 entity binary indicators and T - 1 time
binary indicators, along with an intercept:
Yit = b0 + b1Xit + g2D2i + g + gnDni
+ d2B2t + g + dTBTt + u it,
(10.20)
where b0, b1, g2, c, gn, and d2, c, dT are unknown coefficients.
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10.4!!Regression with Time Fixed Effects
331
When there are additional observed “X” regressors, then these appear in Equations (10.19) and (10.20) as well.
The combined entity and time fixed effects regression model eliminates omitted
variables bias arising both from unobserved variables that are constant over time and
from unobserved variables that are constant across states.
Estimation. The time fixed effects model and the entity and time fixed effects model
are both variants of the multiple regression model. Thus their coefficients can be
estimated by OLS by including the additional time and entity binary variables. Alternatively, in a balanced panel the coefficients on the X’s can be computed by first
deviating Y and the X’s from their entity and time-period means and then by estimating the multiple regression equation of deviated Y on the deviated X’s. This algorithm, which is commonly implemented in regression software, eliminates the need
to construct the full set of binary indicators that appear in Equation (10.20). An
equivalent approach is to deviate Y, the X’s, and the time indicators from their entity
(but not time-period) means and to estimate k + T coefficients by multiple regression of the deviated Y on the deviated X’s and the deviated time indicators. Finally,
if T = 2, the entity and time fixed effects regression can be estimated using the
“before and after” approach of Section 10.2, including the intercept in the regression.
Thus the “before and after” regression reported in Equation (10.8), in which the
change in FatalityRate from 1982 to 1988 is regressed on the change in BeerTax from
1982 to 1988 including an intercept, provides the same estimate of the slope coefficient as the OLS regression of FatalityRate on BeerTax, including entity and time
fixed effects, estimated using data for the two years 1982 and 1988.
Application to traffic deaths. Adding time effects to the state fixed effects regression results in the OLS estimate of the regression line:
FatalityRate = -0.64 BeerTax + State Fixed Effects + Time Fixed Effects. (10.21)
10.362
This specification includes the beer tax, 47 state binary variables (state fixed effects),
6 single-year binary variables (time fixed effects), and an intercept, so this regression
actually has 1 + 47 + 6 + 1 = 55 right-hand variables! The coefficients on the time
and state binary variables and the intercept are not reported because they are not of
primary interest.
Including time effects has little impact on the coefficient on the real beer tax
[compare Equations (10.15) and (10.21)]. Although this coefficient is less precisely
estimated when time effects are included, it is still significant at the 10%, but not the
5%, significance level 1t = -0.64 > 0.36 = -1.782.
This estimated relationship between the real beer tax and traffic fatalities is
immune to omitted variable bias from variables that are constant either over time or
across states. However, many important determinants of traffic deaths do not fall into
this category, so this specification could still be subject to omitted variable bias.
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CHAPTER 10!!Regression with Panel Data
Section 10.6 therefore undertakes a more complete empirical examination of the
effect of the beer tax and of laws aimed directly at eliminating drunk driving, controlling for a variety of factors. Before turning to that study, we first discuss the assumptions underlying panel data regression and the construction of standard errors for
fixed effects estimators.
10.5 The Fixed Effects Regression Assumptions and
Standard Errors for Fixed Effects Regression
In panel data, the regression error can be correlated over time within an entity. Like
heteroskedasticity, this correlation does not introduce bias into the fixed effects estimator, but it affects the variance of the fixed effects estimator, and therefore it affects
how one computes standard errors. The standard errors for fixed effects regressions
reported in this chapter are so-called clustered standard errors, which are robust both
to heteroskedasticity and to correlation over time within an entity. When there are
many entities (when n is large), hypothesis tests and confidence intervals can be
computed using the usual large-sample normal and F critical values.
This section describes clustered standard errors. We begin with the fixed effects
regression assumptions, which extend the least squares regression assumptions for
causal inference to panel data; under these assumptions, the fixed effects estimator
is consistent and asymptotically normally distributed when n is large. To keep the
notation as simple as possible, this section focuses on the entity fixed effects regression model of Section 10.3, in which there are no time effects.
The Fixed Effects Regression Assumptions
The four fixed effects regression assumptions are summarized in Key Concept 10.3.
These assumptions extend the four least squares assumptions for causal inference,
stated for cross-sectional data in Key Concept 6.4, to panel data.
The first assumption is that the error term has conditional mean 0 given all T
values of X for that entity. This assumption plays the same role as the first least
squares assumption for cross-sectional data in Key Concept 6.4 and implies that
there is no omitted variable bias. The requirement that the conditional mean of uit
not depend on any of the values of X for that entity—past, present, or future—adds
an important subtlety beyond the first least squares assumption for cross-sectional
data. This assumption is violated if current uit is correlated with past, present, or
future values of X.
The second assumption is that the variables for one entity are distributed identically to, but independently of, the variables for another entity; that is, the variables
are i.i.d. across entities for i = 1, c, n. Like the second least squares assumption in
Key Concept 6.4, the second assumption for fixed effects regression holds if entities
are selected by simple random sampling from the population.
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10.5!!The Fixed Effects Regression Assumptions and Standard Errors for Fixed Effects Regression
The Fixed Effects Regression Assumptions
Yit = b1Xit + ai + u it, i = 1, c, n, t = 1, c, T,
333
KEY CONCEPT
10.3
where b1 is the causal effect on Y of X and
1. u it has conditional mean 0: E1u it " Xi1, Xi2, c, XiT, ai 2 = 0.
2. 1Xi1, Xi2, c, XiT, u i1, u i2, c, u iT 2, i = 1, c, n, are i.i.d. draws from their
joint distribution.
3. Large outliers are unlikely: 1Xit, u it 2 have nonzero finite fourth moments.
4. There is no perfect multicollinearity.
For multiple regressors, Xit should be replaced by the full list X1,it, X2,it, c, Xk,it.
The third and fourth assumptions for fixed effects regression are analogous to
the third and fourth least squares assumptions for cross-sectional data in Key
Concept 6.4.
Under the least squares assumptions for panel data in Key Concept 10.3, the
fixed effects estimator is consistent and is normally distributed when n is large. The
details are discussed in Appendix 10.2.
An important difference between the panel data assumptions in Key Concept
10.3 and the assumptions for cross-sectional data in Key Concept 6.4 is assumption 2.
The cross-sectional counterpart of assumption 2 holds that each observation is independent, which arises under simple random sampling. In contrast, assumption 2 for
panel data holds that the variables are independent across entities but makes no such
restriction within an entity. For example, assumption 2 allows Xit to be correlated
over time within an entity.
If Xit is correlated with Xis for different values of s and t—that is, if Xit is correlated over time for a given entity—then Xit is said to be autocorrelated (correlated
with itself, at different dates) or serially correlated. Autocorrelation is a pervasive
feature of time series data: What happens one year tends to be correlated with what
happens the next year. In the traffic fatality example, Xit, the beer tax in state i in
year t, is autocorrelated: Most of the time the legislature does not change the beer
tax, so if it is high one year relative to its mean value for state i, it will tend to be high
the next year, too. Similarly, it is possible to think of reasons why u it would be autocorrelated. Recall that u it consists of time-varying factors that are determinants of Yit
but are not included as regressors, and some of these omitted factors might be autocorrelated. For example, a downturn in the local economy might produce layoffs and
diminish commuting traffic, thus reducing traffic fatalities for 2 or more years. Similarly, a major road improvement project might reduce traffic accidents not only in the
year of completion but also in future years. Such omitted factors, which persist over
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CHAPTER 10!!Regression with Panel Data
multiple years, produce autocorrelated regression errors. Not all omitted factors will
produce autocorrelation in u it; for example, severe winter driving conditions plausibly affect fatalities, but if winter weather conditions for a given state are independently distributed from one year to the next, then this component of the error term
would be serially uncorrelated. In general, though, as long as some omitted factors
are autocorrelated, then u it will be autocorrelated.
Standard Errors for Fixed Effects Regression
If the regression errors are autocorrelated, then the usual heteroskedasticity-robust
standard error formula for cross-section regression [Equations (5.3) and (5.4)] is not
valid. One way to see this is to draw an analogy to heteroskedasticity. In a regression
with cross-sectional data, if the errors are heteroskedastic, then (as discussed in
Section 5.4) the homoskedasticity-only standard errors are not valid because they
were derived under the false assumption of homoskedasticity. Similarly, if the errors
are autocorrelated, then the usual standard errors will not be valid because they were
derived under the false assumption of no serial correlation.
Standard errors that are valid if u it is potentially heteroskedastic and potentially
correlated over time within an entity are referred to as heteroskedasticity-and
autocorrelation-robust (HAR) standard errors. The standard errors used in this
chapter are one type of HAR standard errors, clustered standard errors. The term
clustered arises because these standard errors allow the regression errors to have an
arbitrary correlation within a cluster, or grouping, but assume that the regression
errors are uncorrelated across clusters. In the context of panel data, each cluster
consists of an entity. Thus clustered standard errors allow for heteroskedasticity and
for arbitrary autocorrelation within an entity but treat the errors as uncorrelated
across entities. That is, clustered standard errors allow for heteroskedasticity and
autocorrelation in a way that is consistent with the second fixed effects regression
assumption in Key Concept 10.3.
Like heteroskedasticity-robust standard errors in regression with cross-sectional
data, clustered standard errors are valid whether or not there is heteroskedasticity,
autocorrelation, or both. If the number of entities n is large, inference using clustered
standard errors can proceed using the usual large-sample normal critical values for
t-statistics and Fq, ∞ critical values for F-statistics testing q restrictions.
In practice, there can be a large difference between clustered standard errors
and standard errors that do not allow for autocorrelation of uit. For example, the
usual (cross-sectional data) heteroskedasticity-robust standard error for the BeerTax coefficient in Equation (10.21) is 0.25, substantially smaller than the clustered
standard error, 0.36, and the respective t-statistics testing b1 = 0 are -2.51 and
-1.78. The reason we report the clustered standard error is that it allows for serial
correlation of uit within an entity, whereas the usual heteroskedasticity-robust
standard error does not. The formula for clustered standard errors is given in
Appendix 10.2.
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10.6!!Drunk Driving Laws and Traffic Deaths
335
10.6 Drunk Driving Laws and Traffic Deaths
Alcohol taxes are only one way to discourage drinking and driving. States differ in
their punishments for drunk driving, and a state that cracks down on drunk driving
could do so by toughening driving laws as well as raising taxes. If so, omitting these
laws could produce omitted variable bias in the OLS estimator of the effect of real
beer taxes on traffic fatalities, even in regressions with state and time fixed effects. In
addition, because vehicle use depends in part on whether drivers have jobs and
because tax changes can reflect economic conditions (a state budget deficit can lead
to tax hikes), omitting state economic conditions also could result in omitted variable
bias. In this section, we therefore extend the preceding analysis of traffic fatalities to
include other driving laws and economic conditions.
The results are summarized in Table 10.1. The format of the table is the same as
that of the tables of regression results in Chapters 7 through 9: Each column reports
a different regression, and each row reports a coefficient estimate and standard error,
a 95% confidence interval for the coefficients on the policy variables of interest, a
F-statistic and p-value, or other information about the regression.
Column (1) in Table 10.1 presents results for the OLS regression of the fatality
rate on the real beer tax without state and time fixed effects. As in the cross-sectional
regressions for 1982 and 1988 [Equations (10.2) and (10.3)], the coefficient on the
real beer tax is positive (0.36): According to this estimate, increasing beer taxes
increases traffic fatalities! However, the regression in column (2) [reported previously as Equation (10.15)], which includes state fixed effects, suggests that the positive coefficient in regression (1) is the result of omitted variable bias (the coefficient
on the real beer tax is -0.66). The regression R2 jumps from 0.091 to 0.889 when fixed
effects are included; evidently, the state fixed effects account for a large amount of
the variation in the data.
Little changes when time effects are added, as reported in column (3) [reported
previously as Equation (10.21)], except that the beer tax coefficient is now estimated
less precisely. The results in columns (1) through (3) are consistent with the omitted
fixed factors—historical and cultural factors, general road conditions, population
density, attitudes toward drinking and driving, and so forth—being important determinants of the variation in traffic fatalities across states.
The next four regressions in Table 10.1 include additional potential determinants
of fatality rates along with state and time effects. The base specification, reported in
column (4), includes variables related to drunk driving laws plus variables that control for the amount of driving and overall state economic conditions. The first legal
variables are the minimum legal drinking age, represented by three binary variables
for a minimum legal drinking age of 18, 19, and 20 (so the omitted group is a minimum legal drinking age of 21 or older). The other legal variable is the punishment
associated with the first conviction for driving under the influence of alcohol, either
mandatory jail time or mandatory community service (the omitted group is less
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CHAPTER 10!!Regression with Panel Data
TABLE 10.1
Regression Analysis of the Effect of Drunk Driving Laws on Traffic Deaths
Dependent variable: traffic fatality rate (deaths per 10,000).
Regressor
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Beer tax
0.36
(0.05)
-0.66
(0.29)
-0.64
(0.36)
-0.45
(0.30)
-0.69
(0.35)
-0.46
(0.31)
-0.93
(0.34)
[0.26, 0.46]
[ - 1.23, - 0.09]
[ - 1.35, 0.07]
[ - 1.04, 0.14]
[ - 1.38, 0.00]
[ - 1.07, 0.15]
[ - 1.60, - 0.26]
0.03
(0.07)
!0.01
(0.08)
0.04
(0.10)
[ - 0.11, 0.17]
[ - 0.17, 0.15]
[ - 0.16, 0.24]
-0.02
(0.05)
-0.08
(0.07)
-0.07
(0.10)
[ - 0.12, 0.08]
[ - 0.21, 0.06]
[ - 0.26, 0.13]
0.03
(0.05)
-0.10
(0.06)
-0.11
(0.13)
[ - 0.07, 0.13]
[ - 0.21, 0.01]
[ - 0.36, 0.14]
Drinking age 18
0.10
Drinking age 19
Drinking age 20
Drinking age
0.00
(0.02)
[ - 0.05, 0.04]
Mandatory jail or
community service?
0.04
(0.10)
0.09
(0.11)
0.04
(0.10)
0.09
(0.16)
[ - 0.17, 0.25]
[ - 0.14, 0.31]
[ - 0.17, 0.25]
[ - 0.24, 0.42]
Average vehicle miles
per driver
0.008
(0.007)
0.017
(0.011)
0.009
(0.007)
0.124
(0.049)
Unemployment rate
-0.063
(0.013)
-0.063
(0.013)
-0.091
(0.021)
Real income per
capita (logarithm)
1.82
(0.64)
1.79
(0.64)
1.00
(0.68)
1982–88
1982 & 1988
only
Years
1982–88
1982–88
1982–88
1982–88
1982–88
State effects?
no
yes
yes
yes
yes
yes
yes
Time effects?
no
no
yes
yes
yes
yes
yes
Clustered standard
errors?
no
yes
yes
yes
yes
yes
yes
10.12
( 60.001)
3.48
(0.006)
10.28
( 60.001)
37.49
(60.001)
0.35
(0.786)
1.41
(0.253)
F-Statistics and p-Values Testing Exclusion of Groups of Variables
Time effects = 0
4.22
(0.002)
Drinking age
coefficients = 0
29.62
( 60.001)
Unemployment rate,
income per capita = 0
R2
0.091
0.889
0.891
0.926
0.893
0.42
(0.738)
31.96
( 60.001)
25.20
( 60.001)
0.926
0.899
These regressions were estimated using panel data for 48 U.S. states. Regressions (1) through (6) use data for all years 1982
to 1988, and regression (7) uses data from 1982 and 1988 only. The data set is described in Appendix 10.1. Standard errors
are given in parentheses under the coefficients, 95% confidence intervals are given in square brackets under the coefficients, and p-values are given in parentheses under the F-statistics.
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10.6!!Drunk Driving Laws and Traffic Deaths
337
severe punishment). The three measures of driving and economic conditions are
average vehicle miles per driver, the unemployment rate, and the logarithm of real
(1988 dollars) personal income per capita (using the logarithm of income permits the
coefficient to be interpreted in terms of percentage changes of income; see
Section 8.2). The final regression in Table 10.1 follows the “before and after” approach
of Section 10.2 and uses only data from 1982 and 1988; thus regression (7) extends
the regression in Equation (10.8) to include the additional regressors.
The regression in column (4) has four interesting results.
1. Including the additional variables reduces the estimated effect of the beer
tax from -0.64 in column (3) to -0.45 in column (4). One way to evaluate
the magnitude of this coefficient is to imagine a state with an average real
beer tax doubling its tax; because the average real beer tax in these data is
approximately $0.50 per case (in 1988 dollars), this entails increasing the tax
by $0.50 per case. The estimated effect of a $0.50 increase in the beer tax is to
decrease the expected fatality rate by 0.45 * 0.50 = 0.23 deaths per 10,000.
This estimated effect is large: Because the average fatality rate is 2 deaths per
10,000, a reduction of 0.23 corresponds to reducing traffic deaths by nearly
one-eighth. This said, the estimate is quite imprecise: Because the standard
error on this coefficient is 0.30, the 95% confidence interval for this effect
is -0.45 * 0.50 { 1.96 * 0.30 * 0.50 = 1 -0.52, 0.082. This wide 95% confidence interval includes 0, so the hypothesis that the beer tax has no effect
cannot be rejected at the 5% significance level.
2. The minimum legal drinking age is precisely estimated to have a small effect
on traffic fatalities. According to the regression in column (4), the 95% confidence interval for the increase in the fatality rate in a state with a minimum legal drinking age of 18, relative to age 21, is 1 -0.11, 0.172. The joint
hypothesis that the coefficients on the minimum legal drinking age variables
are 0 cannot be rejected at the 10% significance level: The F-statistic testing
the joint hypothesis that the three coefficients are 0 is 0.35, with a p-value
of 0.786.
3. The coefficient on the first offense punishment variable is also estimated to
be small and is not significantly different from 0 at the 10% significance level.
4. The economic variables have considerable explanatory power for traffic fatalities. High unemployment rates are associated with fewer fatalities: An increase
in the unemployment rate by 1 percentage point is estimated to reduce traffic
fatalities by 0.063 deaths per 10,000. Similarly, high values of real per capita income are associated with high fatalities: The coefficient is 1.82, so a 1%
increase in real per capita income is associated with an increase in traffic fatalities of 0.0182 deaths per 10,000 (see case I in Key Concept 8.2 for interpretation
of this coefficient). According to these estimates, good economic conditions
are associated with higher fatalities, perhaps because of increased traffic density when the unemployment rate is low or greater alcohol consumption when
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CHAPTER 10!!Regression with Panel Data
income is high. The two economic variables are jointly significant at the 0.1%
significance level (the F-statistic is 29.62).
Columns (5) through (7) of Table 10.1 report regressions that check the sensitivity of these conclusions to changes in the base specification. The regression in column
(5) drops the variables that control for economic conditions. The result is an increase
in the estimated effect of the real beer tax, which becomes significant at the 5% level,
but there is no appreciable change in the other coefficients. The sensitivity of the
estimated beer tax coefficient to including the economic variables, combined with the
statistical significance of the coefficients on those variables in column (4), indicates
that the economic variables should remain in the base specification. The regression
in column (6) shows that the results in column (4) are not sensitive to changing the
functional form when the three drinking age indicator variables are replaced by
the drinking age itself. When the coefficients are estimated using the changes of the
variables from 1982 to 1988 [column (7)], as in Section 10.2, the findings from column
(4) are largely unchanged except that the coefficient on the beer tax is larger and is
significant at the 1% level.
The strength of this analysis is that including state and time fixed effects mitigates the threat of omitted variable bias arising from unobserved variables that either
do not change over time (like cultural attitudes toward drinking and driving) or do
not vary across states (like safety innovations). As always, however, it is important to
think about possible threats to validity. One potential source of omitted variable bias
is that the measure of alcohol taxes used here, the real tax on beer, could move with
other alcohol taxes, which suggests interpreting the results as pertaining more broadly
than just to beer. A subtler possibility is that hikes in the real beer tax could be associated with public education campaigns. If so, changes in the real beer tax could pick
up the effect of a broader campaign to reduce drunk driving.
Taken together, these results present a provocative picture of measures to control drunk driving and traffic fatalities. According to these estimates, neither stiff
punishments nor increases in the minimum legal drinking age have important effects
on fatalities. In contrast, there is evidence that increasing alcohol taxes, as measured
by the real tax on beer, does reduce traffic deaths, presumably through reduced alcohol consumption. The imprecision of the estimated beer tax coefficient means, however, that we should be cautious about drawing policy conclusions from this analysis
and that additional research is warranted.2
2
For further analysis of these data, see Ruhm (1996). A meta-analysis by Wagenaar, Salois, and Komro
(2009) of 112 studies of the effect of alcohol prices and taxes on consumption found elasticities of - 0.46
for beer, - 0.69 for wine, and - 0.80 for spirits and concluded that alcohol taxes have large effects on
reducing consumption relative to other programs. Carpenter and Dobkin (2011) provide evidence that,
in contrast to the findings here, raising the minimum legal drinking age substantially reduces fatalities
among drivers in the affected age range, especially at night, although they do not control for the other
variables in Table 10.1. To learn more about drunk driving and alcohol and about the economics of alcohol
more generally, also see Cook and Moore (2000), Chaloupka, Grossman, and Saffer (2002), Young and
Bielinska-Kwapisz (2006), and Dang (2008).
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Summary
339
10.7 Conclusion
This chapter showed how multiple observations over time on the same entity can be
used to control for unobserved omitted variables that differ across entities but are
constant over time. The key insight is that if the unobserved variable does not change
over time, then any changes in the dependent variable must be due to influences
other than these fixed characteristics. If cultural attitudes toward drinking and driving do not change appreciably over 7 years within a state, then explanations for
changes in the traffic fatality rate over those 7 years must lie elsewhere.
To exploit this insight, you need data in which the same entity is observed at two
or more time periods; that is, you need panel data. With panel data, the multiple
regression model of Part II can be extended to include a full set of entity binary
variables; this is the fixed effects regression model, which can be estimated by OLS.
A twist on the fixed effects regression model is to include time fixed effects, which
control for unobserved variables that change over time but are constant across entities. Both entity and time fixed effects can be included in the regression to control
for variables that vary across entities but are constant over time and for variables that
vary over time but are constant across entities.
Despite these virtues, entity and time fixed effects regression cannot control for
omitted variables that vary both across entities and over time. And, obviously, panel
data methods require panel data, which often are not available. Thus there remains
a need for a method that can eliminate the influence of unobserved omitted variables
when panel data methods cannot do the job. A powerful and general method for
doing so is instrumental variables regression, the topic of Chapter 12.
Summary
1. Panel data consist of observations on multiple (n) entities—states, firms, people,
and so forth—where each entity is observed at two or more time periods (T).
2. Regression with entity fixed effects controls for unobserved variables that differ from one entity to the next but remain constant over time.
3. When there are two time periods, fixed effects regression can be estimated by
a “before and after” regression of the change in Y from the first period to the
second on the corresponding change in X.
4. Entity fixed effects regression can be estimated by including binary variables
for n - 1 entities plus the observable independent variables (the X’s) and an
intercept.
5. Time fixed effects control for unobserved variables that are the same across
entities but vary over time.
6. A regression with time and entity fixed effects can be estimated by including
binary variables for n - 1 entities and binary variables for T - 1 time periods
plus the X’s and an intercept.
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CHAPTER 10!!Regression with Panel Data
7. In panel data, variables are typically autocorrelated—that is, correlated over
time within an entity. Standard errors need to allow both for this autocorrelation and for potential heteroskedasticity, and one way to do so is to use
clustered standard errors.
Key Terms
panel data (320)
balanced panel (320)
unbalanced panel (320)
fixed effects regression model (326)
entity fixed effects (326)
time fixed effects regression model
(330)
time fixed effects (330)
entity and time fixed effects regression
model (330)
autocorrelated (333)
serially correlated (333)
heteroskedasticity-and
autocorrelation-robust (HAR)
standard errors (334)
clustered standard errors (334)
MyLab Economics Can Help You Get a Better Grade
If your exam were tomorrow, would you be ready? For each
MyLab Economics chapter,
MyLab Economics Practice Tests and Study Plan
help you prepare for your exams. You can also find the Exercises and all Review the Concepts Questions
available now in MyLab Economics. To see how it works, turn to the MyLab Economics spread on the
inside front cover of this text and then go to www.pearson.com/mylab/economics.
For additional Empirical Exercises and Data Sets, log on to the Companion Website at
www.pearsonhighered.com/stock_watson.
Review the Concepts
10.1
Why is it necessary to use two subscripts, i and t, to describe panel data? What
does i refer to? What does t refer to?
10.2
A researcher is using a panel data set on n = 1000 workers over T = 10 years
(from 2008 through 2017) that contains the workers’ earnings, sex, education, and age. The researcher is interested in the effect of education on earnings. Give some examples of unobserved person-specific variables that are
correlated with both education and earnings. Can you think of examples of
time-specific variables that might be correlated with education and earnings?
How would you control for these person-specific and time-specific effects in
a panel data regression?
10.3
Can the regression that you suggested in response to Question 10.2 be used
to estimate the effect of a worker’s sex on his or her earnings? Can that
regression be used to estimate the effect of the national unemployment rate
on an individual’s earnings? Explain.
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Exercises
10.4
341
In the context of the regression you suggested for Question 10.2, explain why
the regression error for a given individual might be serially correlated.
Exercises
10.1
This exercise refers to the drunk driving panel data regressions summarized
in Table 10.1.
a. New Jersey has a population of 8.1 million people. Suppose New Jersey increased the tax on a case of beer by $1 (in 1988 dollars). Use the
results in column (4) to predict the number of lives that would be saved
over the next year. Construct a 95% confidence interval for your answer.
b. The drinking age in New Jersey is 21. Suppose New Jersey lowered its
drinking age to 18. Use the results in column (4) to predict the change in
the number of traffic fatalities in the next year. Construct a 95% confidence interval for your answer.
c. Should time effects be included in the regression? Why or why not?
d. A researcher conjectures that the unemployment rate has a different
effect on traffic fatalities in the western states than in the other states.
How would you test this hypothesis? (Be specific about the specification
of the regression and the statistical test you would use.)
10.2
Consider the binary variable version of the fixed effects model in Equation
(10.11) except with an additional regressor, D1i; that is, let
Yit = b0 + b1Xit + g1D1i + g2D2i + g + gnDni + u it.
a. Suppose that n = 3. Show that the binary regressors and the “constant”
regressor are perfectly multicollinear; that is, express one of the variables
D1i, D2i, D3i, and X0,it as a perfect linear function of the others, where
X0,it = 1 for all i, t.
b. Show the result in (a) for general n.
c. What will happen if you try to estimate the coefficients of the regression
by OLS?
10.3
Section 9.2 gave a list of five potential threats to the internal validity of a
regression study. Apply that list to the empirical analysis in Section 10.6 and
thereby draw conclusions about its internal validity.
10.4
Using the regression in Equation (10.11), what are the slope and intercept for
a. Entity 1 in time period 1?
b. Entity 1 in time period 3?
c. Entity 3 in time period 1?
d. Entity 3 in time period 3?
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CHAPTER 10!!Regression with Panel Data
10.5
Consider the model with a single regressor. This model also can be written as
Yit = b0 + b1X1,it + d2B2t + g + dTBTt + g2D2i + g + gnDni + u it,
10.6
10.7
where B2t = 1 if t = 2 and 0 otherwise, D2i = 1 if i = 2 and 0 otherwise, and
so forth. How are the coefficients 1b0, d2, c, dT, g2, c, gn 2 related to the
coefficients 1a1, c, an, l1, c, lT 2?
Do the fixed effects regression assumptions in Key Concept 10.3 imply that
∼ 2 = 0 for t ≠ s in Equation (10.28)? Explain.
cov1 ∼
vit,v
is
A researcher believes that traffic fatalities increase when roads are icy and
thinks that therefore states with more snow will have more fatalities than
other states. Comment on the following methods designed to estimate the
effect of snow on fatalities:
a. The researcher collects data on the average snowfall for each state and
adds this regressor (AverageSnowi) to the regressions given in Table 10.1.
b. The researcher collects data on the snowfall in each state for each year
in the sample and adds this regressor to the regressions.
10.8
Consider observations 1Yit, Xit 2 from the linear panel data model
Yit = Xitb1 + ai + lit + u it,
where t = 1, c, T; i = 1, c, n; and ai + lit is an unobserved entity-specific
time trend. How would you estimate b1?
10.9
a. In the fixed effects regression model, are the fixed entity effects, ai,
consistently estimated as n ¡ ∞ with T fixed? (Hint: Analyze the
model with no X’s: Yit = ai + u it.)
b. If n is large (say, n = 2000) but T is small (say, T = 4), do you think that
the estimated values of ai are approximately normally distributed? Why
or why not? (Hint: Analyze the model Yit = ai + u it.)
10.10 In a study of the effect on earnings of education using panel data on annual
earnings for a large number of workers, a researcher regresses earnings in a
given year on age, education, union status, and the worker’s earnings in the
previous year, using fixed effects regression. Will this regression give reliable
estimates of the effects of the regressors (age, education, union status, and
previous year’s earnings) on earnings? Explain. (Hint: Check the fixed effects
regression assumptions in Section 10.5.)
10.11 Let bnDM
denote the entity-demeaned estimator given in Equation (10.22),
1
and let bnBA
1 denote the “before and after” estimator without an intercept, so
BA
n
that b1 = 3 Σ ni= 1 1Xi2 - Xi1 2 1Yi2 - Yi1 2 4 > 3 Σ ni= 1 1Xi2 - Xi1 2 2 4. Show that, if
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Empirical Exercises
343
∼
T = 2, bnDM
= bnBA
1
1 . [Hint: Use the definition of Xit before Equation (10.22)
∼
∼
1
1
to show that Xi1 = - 2 1Xi2 - Xi1 2 and Xi2 = 2 1Xi2 - Xi1 2.]
Empirical Exercises
E10.1
Some U.S. states have enacted laws that allow citizens to carry concealed
weapons. These laws are known as “shall-issue” laws because they instruct
local authorities to issue a concealed weapons permit to all applicants who
are citizens, are mentally competent, and have not been convicted of a felony.
(Some states have some additional restrictions.) Proponents argue that if
more people carry concealed weapons, crime will decline because criminals
will be deterred from attacking other people. Opponents argue that crime
will increase because of accidental or spontaneous use of the weapons. In this
exercise, you will analyze the effect of concealed weapons laws on violent
crimes. On the text website, http://www.pearsonhighered.com/stock_watson,
you will find the data file Guns, which contains a balanced panel of data from
the 50 U.S. states plus the District of Columbia for the years 1977 through
1999.3 A detailed description is given in Guns_Description, available on the
website.
a. Estimate (1) a regression of ln(vio) against shall and (2) a regression of
ln(vio) against shall, incarc_rate, density, avginc, pop, pb1064, pw1064,
and pm1029.
i. Interpret the coefficient on shall in regression (2). Is this estimate
large or small in a real-world sense?
ii. Does adding the control variables in regression (2) change the
estimated effect of a shall-issue law in regression (1) as measured
by!statistical significance? As measured by the real-world significance
of the estimated coefficient?
iii. Suggest a variable that varies across states but plausibly varies
little—or not at all—over time and that could cause omitted variable
bias in regression (2).
b. Do the results change when you add fixed state effects? If so, which set
of regression results is more credible, and why?
c. Do the results change when you add fixed time effects? If so, which set
of regression results is more credible, and why?
d. Repeat the analysis using ln(rob) and ln(mur) in place of ln(vio).
3
These data were provided by Professor John Donohue of Stanford University and were used in his paper
with Ian Ayres, “Shooting Down the ‘More Guns Less Crime’ Hypothesis,” Stanford Law Review, 2003,
55: 1193–1312.
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CHAPTER 10!!Regression with Panel Data
e. In your view, what are the most important remaining threats to the
internal validity of this regression analysis?
f. Based on your analysis, what conclusions would you draw about the
effects of concealed weapons laws on these crime rates?
E10.2
Do citizens demand more democracy and political freedom as their incomes
grow? That is, is democracy a normal good? On the text website, http://www
.pearsonhighered.com/stock_watson, you will find the data file Income_
Democracy, which contains a panel data set from 195 countries for the years
1960, 1965, . . . , 2000. A detailed description is given in Income_Democracy_
Description, available on the website.4 The data set contains an index of political freedom/democracy for each country in each year, together with data on
each country’s income and various demographic controls. (The income and
demographic controls are lagged five years relative to the democracy index to
allow time for democracy to adjust to changes in these variables.)
a. Is the data set a balanced panel? Explain.
b. The index of political freedom/democracy is labeled Dem_ind.
i. What are the minimum and maximum values of Dem_ind in the data
set? What are the mean and standard deviation of Dem_ind in the
data set? What are the 10th, 25th, 50th, 75th, and 90th percentiles of
its distribution?
ii. What is the value of Dem_ind for the United States in 2000?
Averaged over all years in the data set?
iii. What is the value of Dem_ind for Libya in 2000? Averaged over all
years in the data set?
iv. List five countries with an average value of Dem_ind greater than
0.95; less than 0.10; and between 0.3 and 0.7.
c. The logarithm of per capita income is labeled Log_GDPPC. Regress
Dem_ind on Log_GDPPC. Use standard errors that are clustered by
country.
i. How large is the estimated coefficient on Log_GDPPC? Is the
coefficient statistically significant?
ii. If per capita income in a country increases by 20%, by how much is
Dem_ind predicted to increase? What is a 95% confidence interval
for the prediction? Is the predicted increase in Dem_ind large or
small? (Explain what you mean by large or small.)
4
These data were provided by Daron Acemoglu of M.I.T. and were used in his paper with Simon Johnson,
James Robinson, and Pierre Yared, “Income and Democracy,” American Economic Review, 2008, 98:3,
808–842.
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The State Traffic Fatality Data Set
345
iii. Why is it important to use clustered standard errors for the regression? Do the results change if you do not use clustered standard
errors?
d.
i. Suggest a variable that varies across countries but plausibly varies
little—or not at all—over time and that could cause omitted variable
bias in the regression in (c).
ii. Estimate the regression in (c), allowing for country fixed effects.
How do your answers to (c)(i) and (c)(ii) change?
iii. Exclude the data for Azerbaijan, and rerun the regression. Do the
results change? Why or why not?
iv. Suggest a variable that varies over time but plausibly varies little—or
not at all—across countries and that could cause omitted variable
bias in the regression in (c).
v. Estimate the regression in (c), allowing for time and country fixed
effects. How do your answers to (c)(i) and (c)(ii) change?
vi. There are additional demographic controls in the data set. Should
these variables be included in the regression? If so, how do the results
change when they are included?
e. Based on your analysis, what conclusions do you draw about the effects
of income on democracy?
APPENDIX
10.1 The State Traffic Fatality Data Set
The data are for the contiguous 48 U.S. states (excluding Alaska and Hawaii), annually for
1982 through 1988. The traffic fatality rate is the number of traffic deaths in a given state in a
given year per 10,000 people living in that state in that year. Traffic fatality data were obtained
from the U.S. Department of Transportation Fatal Accident Reporting System. The beer tax
(the tax on a case of beer) was obtained from Beer Institute’s Brewers Almanac. The drinking
age variables in Table 10.1 are binary variables indicating whether the legal drinking age is
18, 19, or 20. The binary punishment variable in Table 10.1 describes the state’s minimum
sentencing requirements for an initial drunk driving conviction: This variable equals 1 if the
state requires jail time or community service and equals 0 otherwise (a lesser punishment).
Data on the total vehicle miles traveled annually by state were obtained from the Department of Transportation. Personal income data were obtained from the U.S. Bureau of Economic Analysis, and the unemployment rate was obtained from the U.S. Bureau of Labor
Statistics.
These data were graciously provided by Professor Christopher J. Ruhm of the Department of Economics at the University of North Carolina.
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CHAPTER 10!!Regression with Panel Data
APPENDIX
10.2 Standard Errors for Fixed Effects Regression
This appendix provides formulas for clustered standard errors for fixed effects regression with
a single regressor. These formulas are extended to multiple regressors in Exercise 19.15.
The Asymptotic Distribution of the Fixed Effects
Estimator with Large n
The fixed effects estimator. The fixed effects estimator of b1 is the OLS estimator obtained
∼
∼
using the entity-demeaned regression of Equation (10.14), in which Yit is regressed on Xit,
∼
∼
T
T
where Yit = Yit - Yi, Xit = Xit - Xi, Yi = T -1 g t = 1Yit, and Xi = T -1 g t = 1Xit. The formula
∼
∼
for the OLS estimator is obtained by replacing Xi - X by Xit and Yi - Y by Yit in Equation
(4.5) and by replacing the single summations in Equation (4.5) by two summations, one over
entities 1i = 1, c, n2 and one over time periods 1t = 1, c, T2,5 so
∼ ∼
a a Xit Yit
n
T
i = 1t = 1
bn1 = n T
∼2
a a X it
.
(10.22)
i = 1t = 1
The derivation of the sampling distribution of bn1 parallels the derivation in Appendix 4.3 of the
sampling distribution of the OLS estimator with cross-sectional data. First, substitute
∼
∼
Yit = b1Xit + ∼
u it [Equation (10.14)] into the numerator of Equation (10.22) to obtain the panel
data counterpart of Equation (4.28):
bn1 = b1 +
1 n T ∼∼
a Xitu it
nT ia
= 1t = 1
1 n T ∼2
a X it
nT ia
= 1t = 1
.
(10.23)
Next rearrange this expression, and multiply both sides by 2nT to obtain
1 n
hi
n T
A n ia
1 T ∼∼
∼
=1
n∼ = 1
2nT1bn1 - b1 2 =
, where hi =
Xitu it and Q
X 2it.
x
a
a
n X∼
A Tt = 1
nT i = 1 ta
Q
=1
(10.24)
The scaling factor in Equation (10.24), nT, is the total number of observations.
Distribution and standard errors when n is large. In most panel data applications, n is
much larger than T, which motivates approximating sampling distributions by letting n S ∞
5
The double summation is the extension to double subscripts of a single summation:
a a Xit = a a a Xit b
n
T
n
T
i = 1t = 1
i=1
n
t=1
= a 1Xi1 + Xi2 + g + XiT 2
i=1
= 1X11 + X12 + g + X1T 2 + 1X21 + X22 + g + X2T 2 + g + 1Xn1 + Xn2 + g + XnT 2.
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Standard Errors for Fixed Effects Regression
347
while keeping T fixed. Under the fixed effects regression assumptions of Key Concept 10.3,
p
∼
n X∼ ¡
QX∼ = ET -1 a Tt= 1X 2it as n S ∞ . Also, hi is i.i.d. over i = 1, c, n (by assumption
Q
2) with mean 0 (by assumption 1) and variance s2h (which is finite by assumption 3), so by the
n
d
N10, s2h 2. It follows from Equation (10.24) that
central limit theorem, 21 > ng i = 1hi ¡
d
2nT1bn1 - b1 2 ¡
Na0,
s2h
Q2X
b.
(10.25)
From Equation (10.25), the variance of the large-sample distribution of bn1 is
2
1 sh
var1bn1 2 =
.
nT Q2X
(10.26)
The clustered standard error formula replaces the population moments in Equation (10.26)
by their sample counterparts:
where s2hn =
SE1bn1 2 =
2
1 shn
n 2∼ ,
C nT QX
n
n
1
1
1hn i - hn 2 2 =
hn 2,
a
a
n - 1i = 1
n - 1i = 1 i
(10.27)
∼
where hn i = 21>T a Tt= 1Xitun it is the sample counterpart of hi 3hn i is hi in Equation (10.24), with
n
∼
u replaced by the fixed effects regression residual un 4 and hn = 11 > n2 g
hn . The final equalit
i=1 i
it
ity in Equation (10.27) arises because hn = 0, which in turn follows from the residuals and
regressors being uncorrelated [Equation (4.32)]. Note that s2h is just the sample variance of hn i
N
[see Equation (3.7)].
The estimator sN2h is a consistent estimator of s2h as n S ∞, even if there is heteroskedasticity or
autocorrelation (Exercise 18.15); thus the clustered standard error in Equation (10.27) is
heteroskedasticity- and autocorrelation-robust. Because the clustered standard error is consistent, the
t-statistic testing b1 = b1,0 has a standard normal distribution under the null hypothesis as n S ∞ .
All the foregoing results apply if there are multiple regressors. In addition, if n is large,
then the F-statistic testing q restrictions (computed using the clustered variance formula) has
its usual asymptotic Fq, ∞ distribution.
Why isn’t the usual heteroskedasticity-robust estimator of Chapter 5 valid for
panel data? There are two reasons. The most important reason is that the heteroskedasticityrobust estimator of Chapter 5 does not allow for serial correlation within a cluster. Recall that,
for two random variables U and V, var1U + V2 = var1U2 + var1V2 + 2cov1U, V2. The
variance hi in Equation (10.24) therefore can be written as the sum of variances plus covari∼
v = X∼
u ; then
ances. Let ∼
it
it it
var1hi 2 = var a
=
1 T ∼
1
vit b = var1 ∼
vi1 + ∼
vi2 + g + ∼
viT 2
A T ta
T
=1
1
var1 ∼
vi1 2 + var1 ∼
vi2 2 + g + var1 ∼
viT 2
T
+ 2cov1 ∼
v ,∼
v 2 + g + 2cov1 ∼
v
,∼
v 24.
i1
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i2
iT - 1
iT
(10.28)
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CHAPTER 10!!Regression with Panel Data
The heteroskedasticity-robust variance formula of Chapter 5 misses all the covariances in the
final part of Equation (10.28), so if there is serial correlation, the usual heteroskedasticityrobust variance estimator is inconsistent.
The second reason is that if T is small, the estimation of the fixed effects introduces bias
into the heteroskedasticity-robust variance estimator. This problem does not arise in crosssectional regression.
The one case in which the usual heteroskedasticity-robust standard errors can be used
with panel data is with fixed effects regression with T = 2 observations. In this case, fixed
effects regression is equivalent to the differences regression in Section 10.2, and
heteroskedasticity-robust and clustered standard errors are equivalent.
For empirical examples showing the importance of using clustered standard errors in
economic panel data, see Bertrand, Duflo, and Mullainathan (2004).
Extensions: Other applications of clustered standard errors. In some cases, u it might
be correlated across entities. For example, in a study of earnings, suppose the sampling scheme
selects families by simple random sampling, then tracks all siblings within a family. Because
the omitted factors that enter the error term could have common elements for siblings, it is not
reasonable to assume that the errors are independent for siblings (even though they are independent across families).
In the siblings example, families are natural clusters, or groupings, of observations, where
u it is correlated within the cluster but not across clusters. The derivation leading to Equation
(10.27) can be modified to allow for clusters across entities (for example, families) or across
both entities and time, as long as there are many clusters.
Clustered standard errors also apply in some applications with cross-sectional data when
collection schemes other than simple random sampling are used. For example, suppose crosssectional student-level data on test scores and student characteristics are obtained by first
randomly sampling classrooms, then collecting data on all students within a classroom. Because
the classrooms are randomly sampled, errors would be uncorrelated for students from different classrooms. However, the errors might be correlated for students within the same classroom, so clustered standard errors would be appropriate, with the clustering done at the
classroom level.
For additional discussion of clustered standard errors, see Cameron and Miller (2015).
Distribution and Standard Errors When n Is Small
If n is small and T is large, then it remains possible to use clustered standard errors; however,
t-statistics need to be compared with critical values from the t n - 1 tables, and the F-statistic
testing q restrictions needs to be compared to the Fq, n - q critical value multiplied by
1n - 12 > 1n - q2. These distributions are valid under the assumptions in Key Concept 10.3,
plus some additional assumptions on the joint distribution of Xit and u it over time within an
entity. Although the validity of the t-distribution in cross-sectional regression requires normality and homoskedasticity of the regression errors (Section 5.6), neither requirement is
needed to justify using the t-distribution with clustered standard errors in panel data when T
is large.
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Standard Errors for Fixed Effects Regression
349
To see why the clustered t-statistic has a t n - 1 distribution when n is small and T is large,
even if uit is neither normally distributed nor homoskedastic, first note that if T is large, then
under additional assumptions, hi in Equation (10.24) will obey a central limit theorem, so
d
hi ¡ N(0, s2h ). (The additional assumptions required for this result are substantial and
technical, and we defer further discussion of them to our treatment of time series data in
Chapter 15.) Thus, if T is large, then 2nT1bn1 - b1 2 in Equation (10.24) is a scaled average of
the n normal random variables hi. Moreover, the clustered formula sN2h in Equation (10.27) is
the usual formula for the sample variance, and if it could be computed using hi, then
1n - 12s2h / s2h would have a x2n - 1 distribution, so the t-statistic would have a t n - 1 distribution
[see Section 3.6]. Using the residuals to compute hn i and s2h does not change this conclusion.
N
In the case of multiple regressors, analogous reasoning leads to the conclusion that the
F-statistic testing q restrictions, computed using the cluster variance estimator, is distributed
as 1 nn -- q1 2Fq, n - q. [For example, the 5% critical value for this F-statistic when n = 10 and
- 1
q = 4 is 1 10
10 - 4 2 * 4.53 = 6.80, where 4.53 is the 5% critical value from the F4,6 distribution
given in Appendix Table 5B.] Note that, as n increases, the t n - 1 and 1 nn -- q1 2Fq, n - q distribu-
tions approach the usual standard normal and Fq, ∞ distributions.6
If both n and T are small, then, in general, bn1 will not be normally distributed, and clustered standard errors will not provide reliable inference.
6
Not all software implements clustered standard errors using the t n - 1 and 1 nn -- q1 2Fq, n - q distributions that
apply if n is small, so you should check how your software implements and treats clustered standard errors.
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CHAPTER
11
Regression with a Binary
Dependent Variable
T
wo people, identical but for their race, walk into a bank and apply for a mortgage,
a large loan so that each can buy an identical house. Does the bank treat them the
same way? Are they both equally likely to have their mortgage application accepted?
By law, they must receive identical treatment. But whether they actually do is a matter
of great concern among bank regulators.
Loans are made and denied for many legitimate reasons. For example, if the
proposed loan payments take up most or all of the applicant’s monthly income, a loan
officer might justifiably deny the loan. Also, even loan officers are human and they can
make honest mistakes, so the denial of a single minority applicant does not prove
anything about discrimination. Many studies of discrimination thus look for statistical
evidence of discrimination, that is, evidence contained in large data sets showing that
whites and minorities are treated differently.
But how, precisely, should one check for statistical evidence of discrimination
in!the mortgage market? A start is to compare the fraction of minority and white
applicants who were denied a mortgage. In the data examined in this chapter,
gathered from mortgage applications in 1990 in the Boston, Massachusetts, area, 28%
of black applicants were denied mortgages but only 9% of white applicants were
denied. But this comparison does not really answer the question that opened this
chapter because the black applicants and the white applicants were not necessarily
“identical but for their race.” Instead, we need a method for comparing rates of denial,
holding other applicant characteristics constant.
This sounds like a job for multiple regression analysis—and it is, but with a twist.
The twist is that the dependent variable—whether the applicant is denied—is binary.
In!Part II, we regularly used binary variables as regressors, and they caused no
particular problems. But when the dependent variable is binary, things are more
difficult: What does it mean to fit a line to a dependent variable that can take on
only!two values, 0 and 1?
The answer to this question is to interpret the regression function as a conditional
probability. This interpretation is discussed in Section 11.1, and it allows us to apply
the multiple regression models from Part II to binary dependent variables. Section 11.1
goes over this “linear probability model.” But the predicted probability interpretation
also suggests that alternative, nonlinear regression models can do a better job
modeling these probabilities. These methods, called “probit” and “logit” regression, are
discussed in Section 11.2. Section 11.3, which is optional, discusses the method used
to estimate the coefficients of the probit and logit regressions, the method of
350
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11.1 !!Binary Dependent Variables and the Linear Probability Model
351
maximum likelihood estimation. In Section 11.4, we apply these methods to the
Boston mortgage application data set to see whether there is evidence of racial bias
in!mortgage lending.
The binary dependent variable considered in this chapter is an example of a
dependent variable with a limited range; in other words, it is a limited dependent
variable. Models for other types of limited dependent variables—for example,
dependent variables that take on multiple discrete values—are surveyed in
Appendix!11.3.
11.1 Binary Dependent Variables
and the Linear Probability Model
Whether a mortgage application is accepted or denied is one example of a binary
variable. Many other important questions also concern binary outcomes. What is the
effect of a tuition subsidy on an individual’s decision to go to college? What determines whether a teenager takes up smoking? What determines whether a country
receives foreign aid? What determines whether a job applicant is successful? In all
these examples, the outcome of interest is binary: The student does or does not go to
college, the teenager does or does not take up smoking, a country does or does not
receive foreign aid, the applicant does or does not get a job.
This section discusses what distinguishes regression with a binary dependent
variable from regression with a continuous dependent variable and then turns to the
simplest model to use with binary dependent variables, the linear probability model.
Binary Dependent Variables
The application examined in this chapter is whether race is a factor in denying a
mortgage application; the binary dependent variable is whether a mortgage application is denied. The data are a subset of a larger data set compiled by researchers at
the Federal Reserve Bank of Boston under the Home Mortgage Disclosure Act
(HMDA) and relate to mortgage applications filed in the Boston, Massachusetts,
area in 1990. The Boston HMDA data are described in Appendix 11.1.
Mortgage applications are complicated. During the period covered by these data,
the decision to approve a loan application typically was made by a bank loan officer.
The loan officer must assess whether the applicant will make his or her loan payments. One important piece of information is the size of the required loan payments
relative to the applicant’s income. As anyone who has borrowed money knows, it is
much easier to make payments that are 10% of your income than 50%! We therefore
begin by looking at the relationship between two variables: the binary dependent
variable deny, which equals 1 if the mortgage application was denied and equals 0 if
it was accepted, and the continuous variable P/I ratio, which is the ratio of the applicant’s anticipated total monthly loan payments to his or her monthly income.
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CHAPTER 11!!Regression with a Binary Dependent Variable
FIGURE 11.1
Scatterplot of Mortgage Application Denial and the Payment-to-Income Ratio
Mortgage applicants with a
high ratio of debt payments
to income (P/I ratio) are more
likely to have their application denied (deny = 1
if denied; deny = 0 if
approved). The linear probability model uses a straight
line to model the probability
of denial, conditional on the
P/I ratio.
Deny
1.4
1.2
1.0
Mortgage denied
0.8
0.6
Linear probability model
0.4
0.2
0.0
Mortgage approved
–0.2
–0.4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
P/I ratio
Figure 11.1 presents a scatterplot of deny versus P/I ratio for 127 of the 2380
observations in the data set. (The scatterplot is easier to read using this subset of the
data.) This scatterplot looks different from the scatterplots of Part II because the
variable deny is binary. Still, it seems to show a relationship between deny and P/I
ratio: Few applicants with a payment-to-income ratio less than 0.3 have their
application denied, but most applicants with a payment-to-income ratio exceeding
0.4 are denied.
This positive relationship between P/I ratio and deny (the higher the P/I ratio,
the greater the fraction of denials) is summarized in Figure 11.1 by the OLS regression line estimated using these 127 observations. As usual, this line plots the predicted value of deny as a function of the regressor, the payment-to-income ratio. For
example, when P>I ratio = 0.3, the predicted value of deny is 0.20. But what, precisely, does it mean for the predicted value of the binary variable deny to be 0.20?
The key to answering this question—and more generally to understanding
regression with a binary dependent variable—is to interpret the regression as modeling the probability that the dependent variable equals 1. Thus the predicted value of
0.20 is interpreted as meaning that, when P/I ratio is 0.3, the probability of denial is
estimated to be 20%. Said differently, if there were many applications with
P>I ratio = 0.3, then 20% of them would be denied.
This interpretation follows from two facts. First, from Part II, the population regression function is the expected value of Y given the regressors, E1Y ! X1, c, Xk2. Second,
from Section 2.2, if Y is a 0–1 binary variable, its expected value (or mean) is the probability that Y = 1; that is, E1Y2 = 0 * Pr1Y = 02 + 1 * Pr1Y = 12 = Pr1Y = 12.
In the regression context, the expected value is conditional on the value of the
regressors, so the probability is conditional on X. Thus for a binary variable,
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11.1 !!Binary Dependent Variables and the Linear Probability Model
353
E1Y ! X1, c, Xk 2 = Pr1Y = 1! X1, c, Xk 2. In short, for a binary dependent variable, the predicted value from the population regression is the probability that Y = 1
given X.
The linear multiple regression model applied to a binary dependent variable is
called the linear probability model: linear because it is a straight line and probability
model because it models the probability that the dependent variable equals 1 (in our
example, the probability of loan denial).
The Linear Probability Model
The linear probability model is the name for the multiple regression model of Part II
when the dependent variable is binary rather than continuous. Because the dependent
variable Y is binary, the population regression function corresponds to the probability that the dependent variable equals 1 given X. The population coefficient b1 on a
regressor X is the change in the probability that Y = 1 associated with a unit change
ni, computed using the estimated regression
in X. Similarly, the OLS predicted value, Y
function, is the predicted probability that the dependent variable equals 1, and the
OLS estimator bn1 estimates the change in the probability that Y = 1 associated with
a unit change in X.
Almost all of the tools of Part II carry over to the linear probability model. The
coefficients can be estimated by OLS. Ninety-five percent confidence intervals can
be formed as {1.96 standard errors, hypotheses concerning several coefficients can
be tested using the F-statistic discussed in Chapter 7, and interactions between variables can be modeled using the methods of Section 8.3. Because the errors of the
linear probability model are always heteroskedastic (Exercise 11.8), it is essential that
heteroskedasticity-robust standard errors be used for inference.
One tool that does not carry over is the R2. When the dependent variable is continuous, it is possible to imagine a situation in which the R2 equals 1: All the data lie
exactly on the regression line. This is impossible when the dependent variable is
binary unless the regressors are also binary. Accordingly, the R2 is not a particularly
useful statistic here. We return to measures of fit in the next section.
The linear probability model is summarized in Key Concept 11.1.
Application to the Boston HMDA data. The OLS regression of the binary dependent variable, deny, against the payment-to-income ratio, P/I ratio, estimated using
all 2380 observations in our data set is
deny = -0.080 + 0.604 P>I ratio.
10.0322 10.0982
(11.1)
The estimated coefficient on P/I ratio is positive, and the population coefficient is
statistically significantly different from 0 at the 1% level (the t-statistic is 6.13). Thus
applicants with higher debt payments as a fraction of income are more likely to have
their application denied. This coefficient can be used to compute the predicted
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CHAPTER 11!!Regression with a Binary Dependent Variable
KEY CONCEPT
11.1
The Linear Probability Model
The linear probability model is the linear multiple regression model,
Yi = b0 + b1X1i + b2X2i + g + bkXki + u i,
(11.2)
applied to a binary dependent variable Yi. Because Y is binary, E1Y ! X1, X2, c, Xk 2 =
Pr1Y = 1! X1, X2, c, Xk 2, so for the linear probability model,
Pr1Y = 1 ! X1, X2, c, Xk 2 = b0 + b1X1 + b2X2 + g + bkXk.
The regression coefficient b1 is the difference in the probability that Y = 1 associated with a unit difference in X1, holding constant the other regressors, and so
forth for b2, c, bk. The regression coefficients can be estimated by OLS, and the
usual (heteroskedasticity-robust) OLS standard errors can be used for confidence
intervals and hypothesis tests.
change in the probability of denial given a change in the regressor. For example,
according to Equation (11.1), if P/I ratio increases by 0.1, the probability of denial
increases by 0.604 * 0.1 ≅ 0.060—that is, by 6.0 percentage points.
The estimated linear probability model in Equation (11.1) can be used to compute predicted denial probabilities as a function of P/I ratio. For example, if projected
debt payments are 30% of an applicant’s income, P/I ratio is 0.3, and the predicted
value from Equation (11.1) is -0.080 + 0.604 * 0.3 = 0.101. That is, according to
this linear probability model, an applicant whose projected debt payments are 30%
of income has a probability of 10.1% that his or her application will be denied. [This
is different from the probability of 20% based on the regression line in Figure 11.1
because that line was estimated using only 127 of the 2380 observations used to estimate Equation (11.1).]
What is the effect of race on the probability of denial, holding constant the P/I
ratio? To keep things simple, we focus on differences between black applicants and
white applicants. To estimate the effect of race, holding constant P/I ratio, we augment Equation (11.1) with a binary regressor that equals 1 if the applicant is black
and equals 0 if the applicant is white. The estimated linear probability model is
deny = -0.091 + 0.559 P>I ratio + 0.177 black.
10.0292 10.0892
10.0252
(11.3)
The coefficient on black, 0.177, indicates that an African American applicant has a
17.7% higher probability of having a mortgage application denied than a white
applicant, holding constant their payment-to-income ratio. This coefficient is significant at the 1% level (the t-statistic is 7.11).
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11.2 !!Probit and Logit Regression
355
Taken literally, this estimate suggests that there might be racial bias in mortgage
decisions, but such a conclusion would be premature. Although the payment-toincome ratio plays a role in the loan officer’s decision, so do many other factors, such
as the applicant’s earning potential and his or her credit history. If any of these variables is correlated with the regressors black given the P/I ratio, its omission from
Equation (11.3) will cause omitted variable bias. Thus we must defer any conclusions
about discrimination in mortgage lending until we complete the more thorough analysis in Section 11.3.
Shortcomings of the linear probability model. The linearity that makes the linear
probability model easy to use is also its major flaw. Because probabilities cannot
exceed 1, the effect on the probability that Y = 1 of a given change in X must be
nonlinear: Although a change in P/I ratio from 0.3 to 0.4 might have a large effect on
the probability of denial, once P/I ratio is so large that the loan is very likely to be
denied, increasing P/I ratio further will have little effect. In contrast, in the linear probability model, the effect of a given change in P/I ratio is constant, which leads to predicted probabilities in Figure 11.1 that drop below 0 for very low values of P/I ratio
and exceed 1 for high values! But this is nonsense: A probability cannot be less than
0 or greater than 1. This nonsensical feature is an inevitable consequence of the linear
regression. To address this problem, we introduce new nonlinear models specifically
designed for binary dependent variables, the probit and logit regression models.
11.2 Probit and Logit Regression
Probit and logit1 regression are nonlinear regression models specifically designed for
binary dependent variables. Because a regression with a binary dependent variable
Y models the probability that Y = 1, it makes sense to adopt a nonlinear formulation
that forces the predicted values to be between 0 and 1. Because cumulative probability distribution functions (c.d.f.’s) produce probabilities between 0 and 1 (Section 2.1),
they are used in logit and probit regressions. Probit regression uses the standard
normal c.d.f. Logit regression, also called logistic regression, uses the logistic c.d.f.
Probit Regression
Probit regression with a single regressor. The probit regression model with a single
regressor X is
Pr1Y = 1! X2 = Φ1b0 + b1X2,
(11.4)
where Φ is the cumulative standard normal distribution function (tabulated in
Appendix Table 1).
Pronounced pro--bit and lo--jit.
1
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CHAPTER 11!!Regression with a Binary Dependent Variable
For example, suppose that Y is the binary mortgage denial variable (deny), X is
the payment-to-income ratio (P/I ratio), b0 = -2, and b1 = 3. What then is the
probability of denial if P/I ratio = 0.4? According to Equation (11.4), this probability
is Φ1b0 + b1P>I ratio2 = Φ1 -2 + 3P>I ratio2 = Φ1 -2 + 3 * 0.42 = Φ1 -0.82.
According to the cumulative normal distribution table (Appendix Table 1),
Φ1 -0.82 = Pr1Z … -0.82 = 21.2%. That is, when P/I ratio is 0.4, the predicted
probability that the application will be denied is 21.2%, computed using the probit
model with the coefficients b0 = -2 and b1 = 3.
In the probit model, the term b0 + b1X plays the role of “z” in the cumulative
standard normal distribution table in Appendix Table 1. Thus the calculation in the
previous paragraph can, equivalently, be done by first computing the “z-value,”
z = b0 + b1X = -2 + 3 * 0.4 = -0.8, and then looking up the probability in the
tail of the normal distribution to the left of z = -0.8, which is 21.2%.
The probit coefficient b1 in Equation (11.4) is the difference in the z-value associated with a unit difference in X. If b1 is positive, a greater value for X increases the
z-value and thus increases the probability that Y = 1; if b1 is negative, a greater value
for X decreases the probability that Y = 1. Although the effect of X on the z-value
is linear, its effect on the probability is nonlinear. Thus in practice the easiest way to
interpret the coefficients of a probit model is to compute the predicted probability,
or the change in the predicted probability, for one or more values of the regressors.
When there is just one regressor, the predicted probability can be plotted as a function of X.
Figure 11.2 plots the estimated regression function produced by the probit
regression of deny on P/I ratio for the 127 observations in the scatterplot. The
FIGURE 11.2
Probit Model of the Probability of Denial Given P/I Ratio
The probit model uses the
cumulative normal distribution function to model the
probability of denial given
the payment-to-income
ratio or, more generally, to
model Pr1Y = 1 ! X2. Unlike
the linear probability model,
the probit conditional probabilities are always between
0 and 1.
Deny
1.4
1.2
1.0
Mortgage denied
0.8
0.6
Probit model
0.4
0.2
0.0
Mortgage approved
–0.2
–0.4
0.0
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
P/I ratio
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11.2 !!Probit and Logit Regression
357
estimated probit regression function has a stretched “S” shape: It is nearly 0 and flat
for small values of P/I ratio, it turns and increases for intermediate values, and it flattens out again and is nearly 1 for large values. For small values of the payment-toincome ratio, the probability of denial is small. For example, for P>I ratio = 0.2, the
estimated probability of denial based on the estimated probit function in Figure 11.2
is Pr1deny = 1 ! P>I ratio = 0.22 = 2.1%. When P/I ratio = 0.3, the estimated probability of denial is 16.1%. When P/I ratio = 0.4, the probability of denial increases
sharply to 51.9%, and when P/I ratio = 0.6, the denial probability is 98.3%. According to this estimated probit model, for applicants with high payment-to-income ratios,
the probability of denial is nearly 1.
Probit regression with multiple regressors. In all the regression problems we have
studied so far, leaving out a determinant of Y that is correlated with the included
regressors results in omitted variable bias. Probit regression is no exception. In linear
regression, the solution is to include the additional variable as a regressor. This is also
the solution to omitted variable bias in probit regression.
The probit model with multiple regressors extends the single-regressor probit
model by adding regressors to compute the z-value. Accordingly, the probit population regression model with two regressors, X1 and X2, is
Pr1Y = 1 ! X1, X2 2 = Φ1b0 + b1X1 + b2X2 2.
(11.5)
For example, suppose that b0 = -1.6, b1 = 2, and b2 = 0.5. If X1 = 0.4 and X2 = 1,
the z-value is z = -1.6 + 2 * 0.4 + 0.5 * 1 = -0.3. So the probability that Y = 1
given X1 = 0.4 and X2 = 1 is Pr1Y = 1! X1 = 0.4, X2 = 12 = Φ1 -0.32 = 38%.
Effect of a change in X. In general, the regression model can be used to determine
the expected change in Y arising from a change in X. When Y is binary, its conditional
expectation is the conditional probability that it equals 1, so the expected change in
Y arising from a change in X is the change in the probability that Y = 1.
Recall from Section 8.1 that, when the population regression function is a nonlinear function of X, this expected change is estimated in three steps: First, compute the predicted value at the original value of X using the estimated regression
function; next, compute the predicted value at the changed value of X, X + ∆X;
finally, compute the difference between the two predicted values. This procedure
is summarized in Key Concept 8.1. As emphasized in Section 8.1, this method
always works for computing predicted effects of a change in X, no matter how
complicated the nonlinear model. When applied to the probit model, the method
of Key Concept 8.1 yields the estimated effect on the probability that Y = 1 of a
change in X.
The probit regression model, predicted probabilities, and estimated effects are
summarized in Key Concept 11.2.
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CHAPTER 11!!Regression with a Binary Dependent Variable
The Probit Model, Predicted Probabilities,
and Estimated Effects
KEY CONCEPT
11.2
The population probit model with multiple regressors is
Pr1Y = 1 ! X1, X2, c, Xk 2 = Φ1b0 + b1X1 + b2X2 + g + bkXk 2,
(11.6)
where the dependent variable Y is binary, Φ is the cumulative standard normal
distribution function, and X1, X2, and so on are regressors. The model is best interpreted
by computing predicted probabilities and the effect of a change in a regressor.
The predicted probability that Y = 1, given values of X1, X2, c, Xk, is calculated by computing the z-value, z = b0 + b1X1 + b2X2 + g + bkXk, and then
looking up this z-value in the normal distribution table (Appendix Table 1).
The coefficient b1 is the difference in the z-value arising from a unit difference
in X1, holding constant X2, c, Xk.
The effect on the predicted probability of a change in a regressor is computed
by (1) computing the predicted probability for the initial value of the regressor,
(2) computing the predicted probability for the new or changed value of the
regressor, and (3) taking their difference.
Application to the mortgage data. As an illustration, we fit a probit model to the
2380 observations in our data set on mortgage denial (deny) and the payment-toincome ratio (P/I ratio):
Pr 1 deny = 1 0 P>I ratio2 = Φ1 -2.19 + 2.97 P>I ratio2.
10.162 10.472
(11.7)
The estimated coefficients of -2.19 and 2.97 are difficult to interpret because they
affect the probability of denial via the z-value. Indeed, the only things that can be
readily concluded from the estimated probit regression in Equation (11.7) are that
the payment-to-income ratio is positively related to probability of denial (the
coefficient on P/I ratio is positive) and that this relationship is statistically significant
1t = 2.97>0.47 = 6.322.
What is the change in the predicted probability that an application will be denied when
the payment-to-income ratio increases from 0.3 to 0.4? To answer this question, we follow
the procedure in Key Concept 8.1: Compute the probability of denial for P>I ratio = 0.3
and for P>I ratio = 0.4, and then compute the difference. The probability of denial when
P>I ratio = 0.3 is Φ1-2.19 + 2.97 * 0.32 = Φ1 -1.302 = 0.097. The probability of
denial when P>I ratio = 0.4 is Φ1-2.19 + 2.97 * 0.42 = Φ1 -1.002 = 0.159. The estimated change in the probability of denial is 0.159 - 0.097 = 0.062. That is, an
increase in the payment-to-income ratio from 0.3 to 0.4 is associated with an increase
in the probability of denial of 6.2 percentage points, from 9.7% to 15.9%.
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11.2 !!Probit and Logit Regression
359
Because the probit regression function is nonlinear, the effect of a change in X
depends on the starting value of X. For example, if P>I ratio = 0.5, the estimated denial
probability based on Equation (11.7) is Φ1 -2.19 + 2.97 * 0.52 = Φ1 -0.712 = 0.239.
Thus the change in the predicted probability when P/I ratio increases from 0.4 to 0.5
is 0.239 - 0.159, or 8.0 percentage points, larger than the increase of 6.2 percentage
points when P/I ratio increases from 0.3 to 0.4.
What is the effect of race on the probability of mortgage denial, holding constant
the payment-to-income ratio? To estimate this effect, we estimate a probit regression
with both P/I ratio and black as regressors:
Pr 1deny = 1 ! P>I ratio, black2 = Φ1 -2.26 + 2.74 P>I ratio + 0.71 black2. (11.8)
10.162 10.442
10.0832
Again, the values of the coefficients are difficult to interpret, but the sign and statistical significance are not. The coefficient on black is positive, indicating that an African
American applicant has a higher probability of denial than a white applicant, holding
constant their payment-to-income ratio. This coefficient is statistically significant at
the 1% level (the t-statistic on the coefficient multiplying black is 8.55). For a white
applicant with P>I ratio = 0.3, the predicted denial probability is 7.5%, while for a
black applicant with P>I ratio = 0.3, it is 23.3%; the difference in denial probabilities
between these two hypothetical applicants is 15.8 percentage points.
Estimation of the probit coefficients. The probit coefficients reported here were
estimated using the method of maximum likelihood, which produces efficient (minimum variance) estimators in a wide variety of applications, including regression with
a binary dependent variable. The maximum likelihood estimator is consistent and
normally distributed in large samples, so t-statistics and confidence intervals for the
coefficients can be constructed in the usual way.
Regression software for estimating probit models typically uses maximum likelihood estimation, so this is a simple method to apply in practice. Standard errors
produced by such software can be used in the same way as the standard errors of
regression coefficients; for example, a 95% confidence interval for the true probit
coefficient can be constructed as the estimated coefficient {1.96 standard errors.
Similarly, F-statistics computed using maximum likelihood estimators can be used to
test joint hypotheses. Maximum likelihood estimation is discussed further in
Section 11.3, with additional details given in Appendix 11.2.
Logit Regression
The logit regression model. The logit regression model is similar to the probit
regression model except that the cumulative standard normal distribution function Φ
in Equation (11.6) is replaced by the cumulative standard logistic distribution function,
which we denote by F. Logit regression is summarized in Key Concept 11.3. The logistic
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CHAPTER 11!!Regression with a Binary Dependent Variable
KEY CONCEPT
11.3
Logit Regression
The population logit model of the binary dependent variable Y with multiple
regressors is
Pr1Y = 1 ! X1, X2, c, Xk 2 = F1b0 + b1X1 + b2X2 + g + bkXk 2
1
=
.
1 + e -1b0 + b1X1 + b2X2 + g + bkXk2
(11.9)
Logit regression is similar to probit regression except that the cumulative distribution function is different.
cumulative distribution function has a specific functional form, defined in terms of the
exponential function, which is given as the final expression in Equation (11.9).
As with probit, the logit coefficients are best interpreted by computing predicted
probabilities and differences in predicted probabilities.
The coefficients of the logit model can be estimated by maximum likelihood. The
maximum likelihood estimator is consistent and normally distributed in large samples, so
t-statistics and confidence intervals for the coefficients can be constructed in the usual way.
The logit and probit regression functions are similar. This is illustrated in
Figure 11.3, which graphs the probit and logit regression functions for the dependent
variable deny and the single regressor P/I ratio, estimated by maximum likelihood
using the same 127 observations as in Figures 11.1 and 11.2. The differences between
the two functions are small.
FIGURE 11.3
Probit and Logit Models of the Probability of Denial Given P/I Ratio
These logit and probit models produce nearly identical
estimates of the probability
that a mortgage application
will be denied, given the
payment-to-income ratio.
Deny
1.4
1.2
1.0
Mortgage denied
0.8
0.6
0.4
Probit model
Logit model
0.2
0.0
Mortgage approved
–0.2
–0.4
0.0
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
P/I ratio
04/07/18 4:52 PM
11.2 !!Probit and Logit Regression
361
Historically, the main motivation for logit regression was that the logistic cumulative distribution function could be computed faster than the normal cumulative
distribution function. With the advent of more powerful computers, this distinction
is no longer important.
Application to the Boston HMDA data. A logit regression of deny against P/I ratio and
black, using the 2380 observations in the data set, yields the estimated regression function
Pr 1deny = 1 ! P>I ratio, black2 = F 1 -4.13 + 5.37 P>I ratio + 1.27 black2. (11.10)
10.352 10.962
10.152
The coefficient on black is positive and statistically significant at the 1% level (the
t-statistic is 8.47). The predicted denial probability of a white applicant with
P>I ratio = 0.3 is 1> 31 + e -1 - 4.13 + 5.37 * 0.3 + 1.27 * 02 4 = 1> 31 + e 2.52 4 = 0.074, or
7.4%. The predicted denial probability of an African American applicant with
P>I ratio = 0.3 is 1> 31 + e 1.25 4 = 0.222, or 22.2%, so the difference between the
two probabilities is 14.8 percentage points.
Comparing the Linear Probability, Probit,
and Logit Models
All three models—linear probability, probit, and logit—are just approximations to
the unknown population regression function E1Y ! X2 = Pr1Y = 1 ! X2. The linear
probability model is easiest to use and to interpret, but it cannot capture the nonlinear nature of the true population regression function. Probit and logit regressions
model this nonlinearity in the probabilities, but their regression coefficients are more
difficult to interpret. So which should you use in practice?
There is no one right answer, and different researchers use different models.
Probit and logit regressions frequently produce similar results. For example, according to the estimated probit model in Equation (11.8), the difference in denial probabilities between a black applicant and a white applicant with P>I ratio = 0.3 was
estimated to be 15.8 percentage points, whereas the logit estimate of this gap, based
on Equation (11.10), was 14.9 percentage points. For practical purposes, the two estimates are very similar. One way to choose between logit and probit is to pick the
method that is easier to use in your statistical software.
The linear probability model provides the least sensible approximation to the
nonlinear population regression function. Even so, in some data sets there may be
few extreme values of the regressors, in which case the linear probability model still
can provide an adequate approximation. In the denial probability regression in
Equation (11.3), the estimated black/white gap from the linear probability model is
17 .7 percentage points, larger than the probit and logit estimates but still qualitatively
similar. The only way to know this, however, is to estimate both a linear and a nonlinear model and to compare their predicted probabilities.
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CHAPTER 11!!Regression with a Binary Dependent Variable
in the Logit
11.3 Estimation and Inference
2
and Probit Models
The nonlinear models studied in Sections 8.2 and 8.3 are nonlinear functions of the
independent variables but are linear functions of the unknown coefficients (parameters). Consequently, the unknown coefficients of those nonlinear regression functions
can be estimated by OLS. In contrast, the probit and logit regression functions are nonlinear functions of the coefficients. That is, the probit coefficients b0, b1, c, bk in Equation (11.6) appear inside the cumulative standard normal distribution function Φ, and
the logit coefficients in Equation (11.9) appear inside the cumulative standard logistic
distribution function F. Because the population regression function is a nonlinear function of the coefficients b0, b1, c, bk, those coefficients cannot be estimated by OLS.
This section provides an introduction to the standard method for estimation of
probit and logit coefficients, maximum likelihood; additional mathematical details
are given in Appendix 11.2. Because it is built into modern statistical software, maximum likelihood estimation of the probit and logit coefficients is easy in practice. The
theory of maximum likelihood estimation, however, is more complicated than the
theory of least squares. We therefore first discuss another estimation method, nonlinear least squares, before turning to maximum likelihood.
Nonlinear Least Squares Estimation
Nonlinear least squares is a general method for estimating the unknown parameters
of a regression function when, like the probit coefficients, those parameters enter the
population regression function nonlinearly. The nonlinear least squares estimator,
which was introduced in Appendix 8.1, extends the OLS estimator to regression functions that are nonlinear functions of the parameters. Like OLS, nonlinear least
squares finds the values of the parameters that minimize the sum of squared prediction mistakes produced by the model.
To be concrete, consider the nonlinear least squares estimator of the parameters
of the probit model. The conditional expectation of Y given the X’s is
E1Y ! X1, c, Xk 2 = Pr1Y = 1! X1, c, Xk 2 = Φ1b0 + b1X1 + g + bkXk 2. Estimation by nonlinear least squares fits this conditional expectation function, which is
a nonlinear function of the parameters, to the dependent variable. That is, the nonlinear least squares estimator of the probit coefficients is the values of b0, c, bk that
minimize the sum of squared prediction mistakes:
a 3Yi - Φ1b0 + b1X1i + g + bkXki 2 4 .
n
2
i=1
(11.11)
The nonlinear least squares estimator shares two key properties with the OLS estimator in linear regression: It is consistent (the probability that it is close to the true
2
This section contains more advanced material that can be skipped without loss of continuity.
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11.3 !!Estimation and Inference in the Logit and Probit Models
363
value approaches 1 as the sample size gets large), and it is normally distributed in
large samples. There are, however, estimators that have a smaller variance than the
nonlinear least squares estimator; that is, the nonlinear least squares estimator is
inefficient. For this reason, the nonlinear least squares estimator of the probit coefficients is rarely used in practice, and instead the parameters are estimated by maximum likelihood.
Maximum Likelihood Estimation
The likelihood function is the joint probability distribution of the data, treated as a
function of the unknown coefficients. The maximum likelihood estimator (MLE) of
the unknown coefficients consists of the values of the coefficients that maximize the
likelihood function. Because the MLE chooses the unknown coefficients to maximize the likelihood function, which is in turn the joint probability distribution, in
effect the MLE chooses the values of the parameters to maximize the probability of
drawing the data that are actually observed. In this sense, the MLEs are the parameter values “most likely” to have produced the data.
To illustrate maximum likelihood estimation, consider two i.i.d. observations, Y1
and Y2, on a binary dependent variable with no regressors. Thus Y is a Bernoulli
random variable, and the only unknown parameter to estimate is the probability p
that Y = 1, which is also the mean of Y.
To obtain the maximum likelihood estimator, we need an expression for the
likelihood function, which in turn requires an expression for the joint probability
distribution of the data. The joint probability distribution of the two observations Y1
and Y2 is Pr1Y1 = y1,Y2 = y2 2. Because Y1 and Y2 are independently distributed,
the!joint distribution is the product of the individual distributions [Equation (2.24)],
so Pr1Y1 = y1, Y2 = y2 2 = Pr1Y1 = y1 2 Pr1Y2 = y2 2. The Bernoulli distribution
can! be summarized in the formula Pr 1 Y = y2 = py 11 - p 2 1 - y : When y = 1,
Pr1Y = 12 = p1 11 - p2 0 = p, and when y = 0, Pr1Y = 02 = p0 11 - p2 1 = 1 - p.
Thus the joint probability distribution of Y 1 and Y 2 is Pr1Y1 = y1,
Y2 = y2 2 = 3py1 11 - p2 1 - y1 4 * 3py2 11 - p2 1 - y2 4 = p1y1 + y22 11 - p2 2 - 1y1 + y22.
The likelihood function is the joint probability distribution, treated as a function
of the unknown coefficients. For n = 2 i.i.d. observations on Bernoulli random variables, the likelihood function is
f1p; Y1, Y2 2 = p1Y1 + Y22 11 - p2 2 - 1Y1 + Y22.
(11.12)
The maximum likelihood estimator of p is the value of p that maximizes the likelihood function in Equation (11.12). As with all maximization or minimization problems, this can be done by trial and error; that is, you can try different values of p and
compute the likelihood f(p; Y1, Y2) until you are satisfied that you have maximized
this function. In this example, however, maximizing the likelihood function using
calculus produces a simple formula for the MLE: The MLE is pn = 12 1Y1 + Y2 2.
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CHAPTER 11!!Regression with a Binary Dependent Variable
In other words, the MLE of p is just the sample average! In fact, for general n, the
MLE pn of the Bernoulli probability p is the sample average; that is, pn = Y (this is
shown in Appendix 11.2). In this example, the MLE is the usual estimator of p, the
fraction of times Yi = 1 in the sample.
This example is similar to the problem of estimating the unknown coefficients of
the probit and logit regression models. In those models, the success probability p is
not constant but rather depends on X; that is, it is the success probability conditional
on X, which is given in Equation (11.6) for the probit model and Equation (11.9) for
the logit model. Thus the probit and logit likelihood functions are similar to the likelihood function in Equation (11.12) except that the success probability varies from one
observation to the next (because it depends on Xi). Expressions for the probit and
logit likelihood functions are given in Appendix 11.2.
Like the nonlinear least squares estimator, the MLE is consistent and normally
distributed in large samples. Because regression software commonly computes the
MLE of the probit coefficients, this estimator is easy to use in practice. All the estimated probit and logit coefficients reported in this chapter are MLEs.
Statistical inference based on the MLE. Because the MLE is normally distributed in
large samples, statistical inference about the probit and logit coefficients based on
the MLE proceeds in the same way as inference about the linear regression function
coefficients based on the OLS estimator. That is, hypothesis tests are performed using
the t-statistic, and 95% confidence intervals are formed as {1.96 standard errors.
Tests of joint hypotheses on multiple coefficients use the F-statistic in a way similar
to that discussed in Chapter 7 for the linear regression model. All of this is completely analogous to statistical inference in the linear regression model.
An important practical point is that some statistical software reports tests of joint
hypotheses using the F-statistic, while other software uses the chi-squared statistic. The
chi-squared statistic is q * F, where q is the number of restrictions being tested. Because
the F-statistic is, under the null hypothesis, distributed as x2q >q in large samples, q * F
is distributed as x2q in large samples. Because the two approaches differ only in whether
they divide by q, they produce identical inferences, but you need to know which
approach is implemented in your software so that you use the correct critical values.
Measures of Fit
In Section 11.1, it was mentioned that the R2 is a poor measure of fit for the linear
probability model. This is also true for probit and logit regression. Two measures of
fit for models with binary dependent variables are the fraction correctly predicted
and the pseudo-R2. The fraction correctly predicted uses the following rule: If Yi = 1
and the predicted probability exceeds 50% or if Yi = 0 and the predicted probability
is less than 50%, then Yi is said to be correctly predicted. Otherwise, Yi is said to be
incorrectly predicted. The fraction correctly predicted is the fraction of the n observations Y1, c, Yn that are correctly predicted.
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11.4 !!Application to the Boston HMDA Data
365
An advantage of this measure of fit is that it is easy to understand. A
disadvantage is that it does not reflect the quality of the prediction: If Yi = 1, the
observation is treated as correctly predicted whether the predicted probability is
51% or 90%.
The pseudo-R2 measures the fit of the model using the likelihood function.
Because the MLE maximizes the likelihood function, adding another regressor to a
probit or logit model increases the value of the maximized likelihood, just like adding
a regressor necessarily reduces the sum of squared residuals in linear regression by
OLS. This suggests measuring the quality of fit of a probit model by comparing values
of the maximized likelihood function with all the regressors to the value of the likelihood with none. This is, in fact, what the pseudo-R2 does. A formula for the pseudo-R2
is given in Appendix 11.2.
11.4 Application to the Boston HMDA Data
The regressions of the previous two sections indicated that denial rates were higher
for black than white applicants, holding constant their payment-to-income ratio.
Loan officers, however, legitimately weigh many factors when deciding on a mortgage application, and if any of those other factors differ systematically by race, the
estimators considered so far have omitted variable bias.
In this section, we take a closer look at whether there is statistical evidence of
discrimination in the Boston HMDA data. Specifically, our objective is to estimate
the effect of race on the probability of denial, holding constant those applicant characteristics that a loan officer might legally consider when deciding on a mortgage
application.
The most important variables available to loan officers through the mortgage
applications in the Boston HMDA data set are listed in Table 11.1; these are the
variables we will focus on in our empirical models of loan decisions. The first two
variables are direct measures of the financial burden the proposed loan would
place on the applicant, measured in terms of his or her income. The first of these is
the P / I ratio; the second is the ratio of housing-related expenses to income. The
next variable is the size of the loan, relative to the assessed value of the home; if
the loan-to-value ratio is nearly 1, the bank might have trouble recouping the full
amount of the loan if the applicant defaults on the loan and the bank forecloses.
The final three financial variables summarize the applicant’s credit history. If an
applicant has been unreliable paying off debts in the past, the loan officer legitimately might worry about the applicant’s ability or desire to make mortgage payments in the future. The three variables measure different types of credit histories,
which the loan officer might weigh differently. The first concerns consumer credit,
such as credit card debt; the second is previous mortgage payment history; and the
third measures credit problems so severe that they appeared in a public legal
record, such as filing for bankruptcy.
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CHAPTER 11!!Regression with a Binary Dependent Variable
TABLE 11.1
Variables Included in Regression Models of Mortgage Decisions
Variable
Definition
Sample Average
Financial Variables
P/I ratio
Ratio of total monthly debt payments to total monthly income
0.331
housing expense-to-income ratio
Ratio of monthly housing expenses to total monthly income
0.255
loan-to-value ratio
Ratio of size of loan to assessed value of property
0.738
consumer credit score
1 if no “slow” payments or delinquencies
2 if one or two slow payments or delinquencies
3 if more than two slow payments
4 if insufficient credit history for determination
5 if delinquent credit history with payments 60 days overdue
6 if delinquent credit history with payments 90 days overdue
2.1
mortgage credit score
1 if no late mortgage payments
2 if no mortgage payment history
3 if one or two late mortgage payments
4 if more than two late mortgage payments
1.7
public bad credit record
1 if any public record of credit problems (bankruptcy, chargeoffs, collection actions)
0 otherwise
0.074
Additional Applicant Characteristics
denied mortgage insurance
1 if applicant applied for mortgage insurance and was denied,
0 otherwise
0.020
self-employed
1 if self-employed, 0 otherwise
0.116
single
1 if applicant reported being single, 0 otherwise
0.393
high school diploma
1 if applicant graduated from high school, 0 otherwise
0.984
unemployment rate
1989 Massachusetts unemployment rate in the applicant’s industry
3.8
condominium
1 if unit is a condominium, 0 otherwise
0.288
black
1 if applicant is black, 0 if white
0.142
deny
1 if mortgage application denied, 0 otherwise
0.120
Table 11.1 also lists some other variables relevant to the loan officer’s decision.
Sometimes the applicant must apply for private mortgage insurance.3 The loan officer knows whether that application was denied, and that denial would weigh negatively with the loan officer. The next four variables, which concern the applicant’s
employment status, marital status, and educational attainment, as well as the unemployment rate in the applicant’s industry, relate to the prospective ability of the applicant to repay. In the event of foreclosure, characteristics of the property are relevant
as well, and the next variable indicates whether the property is a condominium. The
final two variables in Table 11.1 are whether the applicant is black or white and
3
Mortgage insurance is an insurance policy under which the insurance company makes the monthly payment to the bank if the borrower defaults. During the period of this study, if the loan-to-value ratio exceeds
80%, the applicant typically was required to buy mortgage insurance.
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11.4 !!Application to the Boston HMDA Data
367
whether the application was denied or accepted. In these data, 14.2% of applicants
are black, and 12.0% of applications are denied.
Table 11.2 presents regression results based on these variables. The base specifications, reported in columns (1) through (3), include the financial variables in
Table 11.1 plus the variables indicating whether private mortgage insurance was
denied and whether the applicant is self-employed. In the 1990s, loan officers commonly used thresholds, or cutoff values, for the loan-to-value ratio, so the base specification for that variable uses binary variables for whether the loan-to-value ratio is
high 1 Ú 0.952, medium (between 0.8 and 0.95), or low 1 6 0.8; this case is omitted to
avoid perfect multicollinearity). The regressors in the first three columns are similar
to those in the base specification considered by the Federal Reserve Bank of Boston
researchers in their original analysis of these data.4 The regressions in columns (1)
through (3) differ only in how the denial probability is modeled, using a linear probability model, a logit model, and a probit model, respectively.
Because the coefficients of the logit and probit models in columns (2)–(6) are not
directly interpretable, the table reports standard errors but not confidence intervals.
In addition, because the aim of these regressions is to approximate the loan officers’
decision rule, it is of interest to know whether individual variables—especially the
applicant’s race—enter that decision rule. Thus the table reports, through asterisks,
whether the test that the coefficient is 0 rejects at the 5% or 1% significance level.
Because the regression in column (1) is a linear probability model, its coefficients
are estimated changes in predicted probabilities arising from a unit change in the independent variable. Accordingly, an increase in P>I ratio of 0.1 is estimated to increase
the probability of denial by 4.5 percentage points (the coefficient on P>Iratio in column
(1) is 0.449, and 0.449 * 0.1 ≅ 0.045). Similarly, having a high loan-to-value ratio
increases the probability of denial: A loan-to-value ratio exceeding 95% is associated
with an 18.9 percentage point increase (the coefficient is 0.189) in the denial probability, relative to the omitted case of a loan-to-value ratio less than 80%, holding the other
variables in column (1) constant. Applicants with a poor credit rating also have a more
difficult time getting a loan, all else being constant, although interestingly the coefficient on consumer credit is statistically significant but the coefficient on mortgage
credit is not. Applicants with a public record of credit problems, such as filing for bankruptcy, have much greater difficulty obtaining a loan: All else equal, a public bad credit
record is estimated to increase the probability of denial by 0.197, or 19.7 percentage
points. Being denied private mortgage insurance is estimated to be virtually decisive:
The estimated coefficient of 0.702 means that being denied mortgage insurance
increases your chance of being denied a mortgage by 70.2 percentage points, all else
4
The difference between the regressors in columns (1) through (3) and those in Munnell et al. (1996),
table 2 (1), is that Munnell et al. include additional indicators for the location of the home and the identity
of the lender, data that are not publicly available; an indicator for a multifamily home, which is irrelevant
here because our subset focuses on single-family homes; and net wealth, which we omit because this variable has a few very large positive and negative values and thus risks making the results sensitive to a few
specific outlier observations.
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CHAPTER 11!!Regression with a Binary Dependent Variable
TABLE 11.2
Mortgage Denial Regressions Using the Boston HMDA Data
Dependent variable: deny = 1 if mortgage application is denied, = 0 if accepted; 2380 observations.
Regression Model
LPM
Logit
Probit
Probit
Probit
Probit
Regressor
(1)
(2)
(3)
(4)
(5)
(6)
black
0.084**
(0.023)
0.688**
(0.182)
0.389**
(0.098)
0.371**
(0.099)
0.363**
(0.100)
0.246
(0.448)
P/I ratio
0.449**
(0.114)
4.76**
(1.33)
2.44**
(0.61)
2.46**
(0.60)
2.62**
(0.61)
2.57**
(0.66)
housing expense-to-income ratio
- 0.048
(0.110)
- 0.11
(1.29)
- 0.18
(0.68)
- 0.30
(0.68)
- 0.50
(0.70)
- 0.54
(0.74)
medium loan-to-value ratio
(0.80 … loan-value ratio … 0.95)
0.031*
(0.013)
0.46**
(0.16)
0.21**
(0.08)
0.22**
(0.08)
0.22**
(0.08)
0.22**
(0.08)
high loan-to-value ratio (loan-value
ratio 7 0.95)
0.189**
(0.050)
1.49**
(0.32)
0.79**
(0.18)
0.79**
(0.18)
0.84**
(0.18)
0.79**
(0.18)
consumer credit score
0.031**
(0.005)
0.29**
(0.04)
0.15**
(0.02)
0.16**
(0.02)
0.34**
(0.11)
0.16**
(0.02)
mortgage credit score
0.021
(0.011)
0.28*
(0.14)
0.15*
(0.07)
0.11
(0.08)
0.16
(0.10)
0.11
(0.08)
public bad credit record
0.197**
(0.035)
1.23**
(0.20)
0.70**
(0.12)
0.70**
(0.12)
0.72**
(0.12)
0.70**
(0.12)
denied mortgage insurance
0.702**
(0.045)
4.55**
(0.57)
2.56**
(0.30)
2.59**
(0.29)
2.59**
(0.30)
2.59**
(0.29)
self-employed
0.060**
(0.021)
0.67**
(0.21)
0.36**
(0.11)
0.35**
(0.11)
0.34**
(0.11)
0.35**
(0.11)
single
0.23**
(0.08)
0.23**
(0.08)
0.23**
(0.08)
high school diploma
"0.61**
(0.23)
"0.60*
(0.24)
"0.62**
(0.23)
unemployment rate
0.03
(0.02)
0.03
(0.02)
0.03
(0.02)
"0.05
(0.09)
condominium
black * P/I ratio
"0.58
(1.47)
black * housing expense-to-income
ratio
1.23
(1.69)
additional credit rating indicator
variables
constant
no
no
no
no
yes
no
- 0.183**
(0.028)
- 5.71**
(0.48)
- 3.04**
(0.23)
- 2.57**
(0.34)
- 2.90**
(0.39)
- 2.54**
(0.35)
(continued)
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11.4 !!Application to the Boston HMDA Data
(Table 11.2 continued)
F-Statistics and p-Values Testing Exclusion of Groups of Variables
(1)
(2)
(3)
applicant single; high school diploma;
industry unemployment rate
(4)
(5)
(6)
5.85
( 6 0.001)
5.22
(0.001)
5.79
( 6 0.001)
1.22
(0.291)
additional credit rating indicator
variables
race interactions and black
4.96
(0.002)
race interactions only
0.27
(0.766)
difference in predicted probability
of denial, white vs. black (percentage points)
8.4%
6.0%
7.1%
6.6%
6.3%
6.5%
These regressions were estimated using the n = 2380 observations in the Boston HMDA data set described in Appendix 11.1.
The linear probability model was estimated by OLS, and probit and logit regressions were estimated by maximum likelihood.
Standard errors are given in parentheses under the coefficients, and p-values are given in parentheses under the F-statistics.
The change in predicted probability in the final row was computed for a hypothetical applicant whose values of the regressors,
other than race, equal the sample mean. Individual coefficients are statistically significant at the *5% or **1% level.
equal. Of the nine variables (other than race) in the regression, the coefficients on all
but two are statistically significant at the 5% level, which is consistent with loan officers’ considering many factors when they make their decisions.
The coefficient on black in regression (1) is 0.084, indicating that the difference
in denial probabilities for black and white applicants is 8.4 percentage points, holding
constant the other variables in the regression. This is statistically significant at the 1%
significance level 1t = 3.652.
The logit and probit estimates reported in columns (2) and (3) yield similar conclusions. In the logit and probit regressions, eight of the nine coefficients on variables other
than race are individually statistically significantly different from 0 at the 5% level, and
the coefficient on black is statistically significant at the 1% level. As discussed in
Section 11.2, because these models are nonlinear, specific values of all the regressors
must be chosen to compute the difference in predicted probabilities for white applicants
and black applicants. A conventional way to make this choice is to consider an “average”
applicant who has the sample average values of all the regressors other than race. The
final row in Table 11.2 reports this estimated difference in probabilities, evaluated for
this average applicant. The estimated racial differentials are similar to each other:
8.4 percentage points for the linear probability model [column (1)], 6.0 percentage
points for the logit model [column (2)], and 7.1 percentage points for the probit model
[column (3)]. These estimated race effects and the coefficients on black are less than in
the regressions of the previous sections, in which the only regressors were P/I ratio and
black, indicating that those earlier estimates had omitted variable bias.
The regressions in columns (4) through (6) investigate the sensitivity of the
results in column (3) to changes in the regression specification. Column (4) modifies
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CHAPTER 11!!Regression with a Binary Dependent Variable
column (3) by including additional applicant characteristics. These characteristics
help to predict whether the loan is denied; for example, having at least a high school
diploma reduces the probability of denial (the estimate is negative, and the coefficient is statistically significant at the 1% level). However, controlling for these personal characteristics does not change the estimated coefficient on black or the
estimated difference in denial probabilities (6.6%) in an important way.
Column (5) breaks out the six consumer credit categories and four mortgage
credit categories to test the null hypothesis that these two variables enter linearly;
this regression also adds a variable indicating whether the property is a condominium. The null hypothesis that the credit rating variables enter the expression for the
z-value linearly is not rejected, nor is the condominium indicator significant, at the
5% level. Most importantly, the estimated racial difference in denial probabilities
(6.3%) is essentially the same as in columns (3) and (4).
Column (6) examines whether there are interactions. Are different standards
applied to evaluating the payment-to-income and housing expense-to-income ratios
for black applicants versus white applicants? The answer appears to be no: The interaction terms are not jointly statistically significant at the 5% level. However, race continues to have a significant effect, because the race indicator and the interaction terms are
jointly statistically significant at the 1% level. Again, the estimated racial difference in
denial probabilities (6.5%) is essentially the same as in the other probit regressions.
In all six specifications, the effect of race on the denial probability, holding other
applicant characteristics constant, is statistically significant at the 1% level. The estimated difference in denial probabilities between black applicants and white applicants ranges from 6.0 percentage points to 8.4 percentage points.
One way to assess whether this differential is large or small is to return to a variation
on the question posed at the beginning of this chapter. Suppose two individuals apply for
a mortgage, one white and one black, but otherwise having the same values of the other
independent variables in regression (3); specifically, aside from race, the values of the other
variables in regression (3) are the sample average values in the HMDA data set. The white
applicant faces a 7.4% chance of denial, but the black applicant faces a 14.5% chance of
denial. The estimated racial difference in denial probabilities, 7.1 percentage points, means
that the black applicant is nearly twice as likely to be denied as the white applicant.
The results in Table 11.2 (and in the original Boston Fed study) provide statistical
evidence of racial patterns in mortgage denial that, by law, ought not be there. This
evidence played an important role in spurring policy changes by bank regulators.5
But economists love a good argument, and not surprisingly these results have also
stimulated a vigorous debate.
Because the suggestion that there is (or was) racial discrimination in lending is
charged, we briefly review some points of this debate. In so doing, it is useful to adopt
the framework of Chapter 9—that is, to consider the internal and external validity of
5
These policy shifts include changes in the way that fair lending examinations were done by federal bank
regulators, changes in inquiries made by the U.S. Department of Justice, and enhanced education programs
for banks and other home loan origination companies.
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11.5 !!Conclusion
371
the results in Table 11.2, which are representative of previous analyses of the
Boston HMDA data. A number of the criticisms made of the original Federal Reserve
Bank of Boston study concern internal validity: possible errors in the data, alternative
nonlinear functional forms, additional interactions, and so forth. The original data
were subjected to a careful audit, some errors were found, and the results reported
here (and in the final published Boston Fed study) are based on the “cleaned” data
set. Estimation of other specifications—different functional forms and/or additional
regressors—also produces estimates of racial differentials comparable to those in
Table 11.2. A potentially more difficult issue of internal validity is whether there is
relevant nonracial financial information obtained during in-person loan interviews,
but not recorded on the loan application itself, that is correlated with race; if so, there
still might be omitted variable bias in the Table 11.2 regressions. Finally, some have
questioned external validity: Even if there was racial discrimination in Boston in 1990,
it is wrong to implicate lenders elsewhere today. Moreover, racial discrimination might
be less likely using modern online applications because the mortgage can be approved
or denied without a face-to-face meeting. The only way to resolve the question of
external validity is to consider data from other locations and years.6
11.5 Conclusion
When the dependent variable Y is binary, the population regression function is the
probability that Y = 1, conditional on the regressors. Estimation of this population
regression function entails finding a functional form that does justice to its probability interpretation, estimating the unknown parameters of that function, and interpreting the results. The resulting predicted values are predicted probabilities, and the
estimated effect of a change in a regressor X is the estimated change in the probability that Y = 1 arising from the change in X.
A natural way to model the probability that Y = 1 given the regressors is to use a
cumulative distribution function, where the argument of the c.d.f. depends on the regressors. Probit regression uses a normal c.d.f. as the regression function, and logit regression
uses a logistic c.d.f. Because these models are nonlinear functions of the unknown
parameters, those parameters are more complicated to estimate than linear regression
coefficients. The standard estimation method is maximum likelihood. In practice, statistical inference using the maximum likelihood estimates proceeds the same way as it
does in linear multiple regression; for example, 95% confidence intervals for a coefficient are constructed as the estimated coefficient {1.96 standard errors.
6
If you are interested in further reading on this topic, a good place to start is the symposium on racial
discrimination and economics in the Spring 1998 issue of the Journal of Economic Perspectives. The article
in that symposium by Helen Ladd (1998) surveys the evidence and debate on racial discrimination in
mortgage lending. A more detailed treatment is given in Goering and Wienk (1996). The U.S. mortgage
market has changed dramatically since the Boston Fed study, including a relaxation of lending standards,
a bubble in housing prices, the financial crisis of 2008–2009, and a return to tighter lending standards. For
an introduction to changes in mortgage markets, see Green and Wachter (2008).
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372
CHAPTER 11!!Regression with a Binary Dependent Variable
James Heckman and Daniel McFadden, Nobel Laureates
T
he 2000 Nobel Prize in Economics was
McFadden was awarded the prize for develop-
awarded jointly to two econometricians, James
ing models for analyzing discrete choice data (does
J. Heckman of the University of Chicago and Dan-
a high school graduate join the military, go to col-
iel L. McFadden of the University of California at
lege, or get a job?). He started by considering the
Berkeley, for fundamental contributions to the anal-
problem of an individual maximizing the expected
ysis of data on individuals and firms. Much of their
utility of each possible choice, which could depend
work addressed difficulties that arise with limited
on observable variables (such as wages, job charac-
dependent variables.
teristics, and family background). He then derived
Heckman was awarded the prize for develop-
models for the individual choice probabilities with
ing tools for handling sample selection. As discussed
unknown coefficients, which in turn could be esti-
in Section 9.2, sample selection bias occurs when the
mated by maximum likelihood. These models and
availability of data is influenced by a selection process
their extensions have proven widely useful in ana-
related to the value of the dependent variable. For
lyzing discrete choice data in many fields, including
example, suppose you want to estimate the relationship
labor economics, health economics, and transporta-
between earnings and some regressor, X, using a ran-
tion economics.
dom sample from the population. If you estimate the
For more information on these and other Nobel
regression using the subsample of employed workers—
laureates in economics, visit the Nobel Foundation
that is, those reporting positive earnings—the OLS
website, http://www.nobel.se/economics.
estimate could be subject to selection bias. Heckman’s
worker is in or out of the labor force (in or out of the
subsample) and to treat this equation and the earnings equation as a system of simultaneous equations.
This general strategy has been extended to selection
problems that arise in many fields, ranging from labor
economics to industrial organization to finance.
Paul Sakuma/AP Images
a binary dependent variable indicating whether the
Henrik Montgomery/
Pressens Bild/AP Images
solution was to specify a preliminary equation with
James J. Heckman
Daniel L. McFadden
Despite its intrinsic nonlinearity, sometimes the population regression function
can be adequately approximated by a linear probability model—that is, by the
straight line produced by linear multiple regression. The linear probability model,
probit regression, and logit regression all give similar bottom-line answers when they
are applied to the Boston HMDA data: All three methods estimate substantial differences in mortgage denial rates for otherwise similar black applicants and white
applicants.
Binary dependent variables are the most common example of limited dependent
variables, which are dependent variables with a limited range. The final quarter of the
20th century saw important advances in econometric methods for analyzing other
limited dependent variables (see the box “James Heckman and Daniel McFadden,
Nobel Laureates”). Some of these methods are reviewed in Appendix 11.3.
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Review the Concepts
373
Summary
1. When Y is a binary variable, the population regression function shows the
probability that Y = 1 given the value of the regressors, X1, X2, c, Xk.
2. The linear multiple regression model is called the linear probability model
when Y is a binary variable because the probability that Y = 1 is a linear function of the regressors.
3. Probit and logit regression models are nonlinear regression models used when
Y is a binary variable. Unlike the linear probability model, probit and logit
regressions ensure that the predicted probability that Y = 1 is between 0 and
1 for all values of X.
4. Probit regression uses the standard normal cumulative distribution function.
Logit regression uses the logistic cumulative distribution function. Logit and
probit coefficients are estimated by maximum likelihood.
5. The values of coefficients in probit and logit regressions are not easy to interpret. Changes in the probability that Y = 1 associated with changes in one or
more of the X’s can be calculated using the general procedure for nonlinear
models outlined in Key Concept 8.1.
6. Hypothesis tests on coefficients in the linear probability, logit, and probit models are performed using the usual t- and F-statistics.
Key Terms
limited dependent variable (351)
linear probability model (353)
probit (355)
logit (355)
logistic regression (355)
likelihood function (363)
maximum likelihood estimator
(MLE) (363)
fraction correctly predicted (364)
pseudo-R2 (365)
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For additional Empirical Exercises and Data Sets, log on to the Companion Website at
www.pearsonhighered.com/stock_watson.
Review the Concepts
11.1
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Suppose a linear probability model yields a predicted value of Y that is equal
to 1.3. Explain why this is nonsensical.
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CHAPTER 11!!Regression with a Binary Dependent Variable
11.2
In Table 11.2, the estimated coefficient on black is 0.084 in column (1), 0.688
in column (2), and 0.389 in column (3). In spite of these large differences, all
three models yield similar estimates of the marginal effect of race on the probability of mortgage denial. How can this be?
11.3
One of your friends is using data on individuals to study the determinants
of smoking at your university. She asks you whether she should use a probit,
logit, or linear probability model. What advice do you give her? Why?
11.4
Why are the coefficients of probit and logit models estimated by maximum
likelihood instead of OLS?
Exercises
Exercises 11.1 through 11.5 are based on the following scenario: Four hundred driver’s license applicants were randomly selected and asked whether they passed their
driving test 1Passi = 12 or failed their test 1Passi = 02; data were also collected on
their sex 1Malei = 1 if male and = 0 if female) and their years of driving experience
(Experiencei, in years). The following table summarizes several estimated models.
11.1
Using the results in column (1):
a. Does the probability of passing the test depend on Experience? Explain.
b. Matthew has 10 years of driving experience. What is the probability that
he will pass the test?
c. Christopher is a new driver (0 years of experience). What is the probability that he will pass the test?
d. The sample included values of Experience between 0 and 40 years, and
only four people in the sample had more than 30 years of driving experience. Jed is 95 years old and has been driving since he was 15. What
Dependent Variable: Pass
Linear
Experience
Linear
Probit
Logit
Probability
Probit
Logit
Probability
Probit
(1)
(2)
(3)
(4)
(5)
(6)
(7)
0.031
(0.009)
0.040
(0.016)
0.006
(0.002)
0.041
(0.156)
- 0.333
(0.161)
Male
- 0.622
(0.303)
- 0.071
(0.034)
Male * Experience
Constant
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- 0.174
(0.259)
- 0.015
(0.019)
0.712
(0.126)
1.059
(0.221)
0.774
(0.034)
1.282
(0.124)
2.197
(0.242)
0.900
(0.022)
0.806
(0.200)
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Exercises
375
is the model’s prediction for the probability that Jed will pass the test?
Do!you think that this prediction is reliable? Why or why not?
11.2
a.! Answer (a) through (c) from Exercise 11.1 using the results in column (2).
b.! Sketch the predicted probabilities from the probit and logit regressions in
columns (1) and (2) for values of Experience between 0 and 60. Are the
probit and logit models similar?
11.3
a.! Answer (a) through (c) from Exercise 11.1 using the results in column (3).
b.! Sketch the predicted probabilities from the probit and linear probability
model regressions in columns (1) and (3) as a function of Experience for
values of Experience between 0 and 60. Do you think that the linear probability is appropriate here? Why or why not?
11.4
Using the results in columns (4) through (6):
a. Compute the estimated probabilities of passing the test for men and for
women.
b. Are the models in (4) through (6) different? Why or why not?
11.5
Using the results in column (7):
a. Akira is a man with 10 years of driving experience. What is the
probability that he will pass the test?
b. Jane is a woman with 2 years of driving experience. What is the
probability that she will pass the test?
c. Does the effect of experience on test performance depend on his or her
sex? Explain.
11.6
Use the estimated probit model in Equation (11.8) to answer the following
questions:
a. A black mortgage applicant has a P/I ratio of 0.35. What is the probability that his application will be denied?
b. Suppose the applicant reduced this ratio to 0.30. What effect would this
have on his probability of being denied a mortgage?
c. Repeat (a) and (b) for a white applicant.
d. Does the marginal effect of the P/I ratio on the probability of mortgage
denial depend on race? Explain.
11.7
Repeat Exercise 11.6 using the logit model in Equation (11.10). Are the logit
and probit results similar? Explain.
11.8
Consider the linear probability model Yi = b0 + b1Xi + u i, and assume that
E1u i ! Xi 2 = 0.
a. Show that Pr1Yi = 1 ! Xi 2 = b0 + b1Xi.
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CHAPTER 11!!Regression with a Binary Dependent Variable
b. Show that var1u i ! Xi 2 = 1b0 + b1Xi 2 31 - 1b0 + b1Xi 2 4. [Hint: Review
Equation (2.7).]
c. Is ui heteroskedastic? Explain.
d. (Requires Section 11.3) Derive the likelihood function.
11.9
Use the estimated linear probability model shown in column (1) of Table 11.2
to answer the following:
a. Two applicants, one white and one black, apply for a mortgage. They
have the same values for all the regressors other than race. How much
more likely is the black applicant to be denied a mortgage?
b. Construct a 95% confidence interval for your answer to (a).
c. Think of an important omitted variable that might bias the answer in (a).
What is it, and how would it bias the results?
11.10 (Requires Section 11.3 and calculus) Suppose a random variable Y has the
following probability distribution: Pr1Y = 12 = p, Pr1Y = 22 = q, and
Pr1Y = 32 = 1 - p - q. A random sample of size n is drawn from this distribution, and the random variables are denoted Y1, Y2, c ,Yn.
a. Derive the likelihood function for the parameters p and q.
b. Derive formulas for the MLE of p and q.
11.11 (Requires Appendix 11.3) Which model would you use for
a. A study explaining the number of minutes that a person spends talking
on a cell phone during the month?
b. A study explaining grades (A through F) in a large Principles of
Economics class?
c. A study of consumers’ choices for Coke, Pepsi, or generic cola?
d. A study of the number of cell phones owned by a family?
Empirical Exercises
E11.1
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In April 2008, the unemployment rate in the United States stood at 5.0%. By
April 2009, it had increased to 9.0%, and it had increased further, to 10.0%, by
October 2009. Were some groups of workers more likely to lose their jobs than
others during the Great Recession? For example, were young workers more
likely to lose their jobs than middle-aged workers? What about workers with
a college degree versus those without a degree or women versus men? On the
text website, http://www.pearsonhighered.com/stock_watson, you will find the
data file Employment_08_09, which contains a random sample of 5440 workers who were surveyed in April 2008 and reported that they were employed
full-time. A detailed description is given in Employment_08_09_Description,
21/08/18 12:23 PM
Empirical Exercises
377
available on the website. These workers were surveyed one year later, in
April 2009, and asked about their employment status (employed, unemployed, or
out of the labor force). The data set also includes various demographic measures
for each individual. Use these data to answer the following questions.
a. What fraction of workers in the sample were employed in April 2009?
Use your answer to compute a 95% confidence interval for the probability that a worker was employed in April 2009, conditional on being
employed in April 2008.
b. Regress Employed on Age and Age2, using a linear probability model.
i. Based on this regression, was age a statistically significant determinant of employment in April 2009?
ii. Is there evidence of a nonlinear effect of age on the probability of
being employed?
iii. Compute the predicted probability of employment for a 20-year-old
worker, a 40-year-old worker, and a 60-year-old worker.
c. Repeat (b) using a probit regression.
d. Repeat (b) using a logit regression.
e. Are there important differences in your answers to (b)–(d)? Explain.
f. The data set includes variables measuring the workers’ educational
attainment, sex, race, marital status, region of the country, and weekly
earnings in April 2008.
i. Construct a table like Table 11.2 to investigate whether the conclusions on the effect of age on employment from (b)–(d) are affected
by omitted variable bias.
ii. Use the regressions in your table to discuss the characteristics of
workers who were hurt most by the Great Recession.
g. The results in (a)–(f) were based on the probability of employment.
Workers who are not employed can either be (i) unemployed or
(ii) out the labor force. Do the conclusions you reached in (a)–(f) also
hold for workers who became unemployed? (Hint: Use the binary
variable Unemployed instead of Employed.)
h. These results have covered employment transitions during the Great
Recession, but what about transitions during normal times? On the text
website, you will find the data file Employment_06_07, which measures
the same variables but for the years 2006–2007. Analyze these data and
comment on the differences in employment transitions during recessions
and normal times.
E11.2
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Believe it or not, workers used to be able to smoke inside office buildings.
Smoking bans were introduced in several areas during the 1990s. Supporters of
these bans argued that in addition to eliminating the externality of secondhand
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CHAPTER 11!!Regression with a Binary Dependent Variable
smoke, they would encourage smokers to quit by reducing their opportunities
to smoke. In this assignment, you will estimate the effect of workplace smoking bans on smoking, using data on a sample of 10,000 U.S. indoor workers
from 1991 to 1993, available on the text website, http://www.pearsonhighered
.com/stock_watson, in the file Smoking. The data set contains information on
whether individuals were or were not subject to a workplace smoking ban,
whether the individuals smoked, and other individual characteristics.7 A
detailed description is given in Smoking_Description, available on the website.
a. Estimate the probability of smoking for (i) all workers, (ii) workers
affected by workplace smoking bans, and (iii) workers not affected by
workplace smoking bans.
b. What is the difference in the probability of smoking between workers
affected by a workplace smoking ban and workers not affected by a
workplace smoking ban? Use a linear probability model to determine
whether this difference is statistically significant.
c. Estimate a linear probability model with smoker as the dependent
variable and the following regressors: smkban, female, age, age2,
hsdrop, hsgrad, colsome, colgrad, black, and hispanic. Compare the
estimated effect of a smoking ban from this regression with your answer
from (b). Suggest an explanation, based on the substance of this regression,
for the change in the estimated effect of a smoking ban between (b) and (c).
d. Test the hypothesis that the coefficient on smkban is 0 in the population
version of the regression in (c) against the alternative that it is nonzero,
at the 5% significance level.
e. Test the hypothesis that the probability of smoking does not depend on
the level of education in the regression in (c). Does the probability of
smoking increase or decrease with the level of education?
f. Repeat (c)–(e) using a probit model.
g. Repeat (c)–(e) using a logit model.
h.
i. Mr. A is white, non-Hispanic, 20 years old, and a high school dropout.
Using the probit regression and assuming that Mr. A is not subject
to a workplace smoking ban, calculate the probability that Mr. A
smokes. Carry out the calculation again, assuming that he is subject
to a workplace smoking ban. What is the effect of the smoking ban
on the probability of smoking?
ii. Repeat (i) for Ms. B, a female, black, 40-year-old college graduate.
iii. Repeat (i)–(ii) using the linear probability model.
7
These data were provided by Professor William Evans of the University of Maryland and were used
in his paper with Matthew Farrelly and Edward Montgomery, “Do Workplace Smoking Bans Reduce
Smoking?” American Economic Review, 1999, 89(4): 728–747.
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Maximum Likelihood Estimation
379
iv. Repeat (i)–(ii) using the logit model.
v. Based on your answers to (i)–(iv), do the logit, probit, and linear
probability models differ? If they do, which results make most sense?
Are the estimated effects large in a real-world sense?
APPENDIX
11.1 The Boston HMDA Data Set
The Boston HMDA data set was collected by researchers at the Federal Reserve Bank of
Boston. The data set combines information from mortgage applications and a follow-up survey
of the banks and other lending institutions that received these mortgage applications. The data
pertain to mortgage applications made in 1990 in the greater Boston metropolitan area. The
full data set has 2925 observations, consisting of all mortgage applications by blacks and Hispanics plus a random sample of mortgage applications by whites.
To narrow the scope of the analysis in this chapter, we use a subset of the data for singlefamily residences only (thereby excluding data on multifamily homes) and for black applicants
and white applicants only (thereby excluding data on applicants from other minority groups). This
leaves 2380 observations. Definitions of the variables used in this chapter are given in Table 11.1.
These data were graciously provided to us by Geoffrey Tootell of the Research Department of the Federal Reserve Bank of Boston. More information about this data set, along with
the conclusions reached by the Federal Reserve Bank of Boston researchers, is available in
Munnell et al. (1996).
APPENDIX
11.2 Maximum Likelihood Estimation
This appendix provides a brief introduction to maximum likelihood estimation in the context
of the binary response models discussed in this chapter. We start by deriving the MLE of the
success probability p for n i.i.d. observations of a Bernoulli random variable. We then turn to
the probit and logit models and discuss the pseudo-R2. We conclude with a discussion of standard errors for predicted probabilities. This appendix uses calculus at two points.
MLE for n i.i.d. Bernoulli Random Variables
The first step in computing the MLE is to derive the joint probability distribution. For n i.i.d.
observations on a Bernoulli random variable, this joint probability distribution is the extension
of the n = 2 case in Section 11.3 to general n:
Pr1Y1 = y1, Y2 = y2, c,Yn = yn 2
= 3py1 11 - p2 11 - y12 4 * 3py2 11 - p2 11 - y22 4 * g * 3pyn 11 - p2 11 - yn2 4
= p1y1 + g + yn2 11 - p2 n - 1y1 + g + yn2.
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(11.13)
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CHAPTER 11!!Regression with a Binary Dependent Variable
coefficients. Let S = g i = 1Yi; then the likelihood function is
The likelihood function is the joint probability distribution, treated as a function of the unknown
n
fBernoulli 1p; Y1, c, Yn 2 = pS 11 - p2 n - S.
(11.14)
The MLE of p is the value of p that maximizes the likelihood in Equation (11.14). The likelihood
function can be maximized using calculus. It is convenient to maximize not the likelihood but
rather its logarithm (because the logarithm is a strictly increasing function, maximizing the
likelihood or its logarithm gives the same estimator). The log likelihood is
Sln1p2 + 1n - S2ln11 - p2, and the derivative of the log likelihood with respect to p is
d
S
n - S
ln 3fBernoulli 1p; Y1, c, Yn 24 =
.
dp
p
1 - p
(11.15)
Setting the derivative in Equation (11.15) to 0 and solving for p yields the MLE pn = S>n = Y.
MLE for the Probit Model
For the probit model, the probability that Yi = 1, conditional on X1i, c, Xki , is
pi = Φ1b0 + b1X1i + g + bkXki 2. The conditional probability distribution for the ith obser-
vation is Pr3Yi = yi ! X1i, c, Xki 4 = pyi i 11 - pi 2 1 - yi. Assuming that 1X1i, c, Xki, Yi 2 are i.i.d.,
i = 1, c, n, the joint probability distribution of Y1, c, Yn, conditional on the X’s, is
Pr1Y1 = y1, c, Yn = yn ! X1i, c, Xki, i = 1, c, n2
= Pr1Y1 = y1 ! X11, c, Xk1 2 * g * Pr1Yn = yn ! X1n, c, Xkn 2
= py11 11 - p1 2 1 - y1 * g * pynn 11 - pn 2 1 - yn.
(11.16)
The likelihood function is the joint probability distribution, treated as a function of the
unknown coefficients. It is conventional to consider the logarithm of the likelihood. Accordingly, the log likelihood function is
ln3fprobit 1b0, c, bk; Y1, c, Yn X1i, c, Xki, i = 1, c, n24
= a Yi ln3Φ1b0 + b1X1i + g + bkXki 24
n
i=1
+ a 11 - Yi 2ln31 - Φ1b0 + b1X1i + g + bk Xki 24,
n
i=1
(11.17)
where this expression incorporates the probit formula for the conditional probability,
pi = Φ1b0 + b1X1i + g + bkXki 2.
The MLE for the probit model maximizes the likelihood function or, equivalently, the
logarithm of the likelihood function given in Equation (11.17). Because there is no simple
formula for the MLE, the probit likelihood function must be maximized using a numerical
algorithm on the computer.
Under general conditions, maximum likelihood estimators are consistent and have a normal sampling distribution in large samples.
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Maximum Likelihood Estimation
MLE for the Logit Model
The likelihood for the logit model is derived in the same way as the likelihood for the probit
model. The only difference is that the conditional success probability pi for the logit model is
given by Equation (11.9). Accordingly, the log likelihood of the logit model is given by Equation (11.17), with Φ1b0 + b1X1i + g + bkXki 2 replaced by 31 + e -1b0 + b1X1i + b2X2i + g + bkXki2 4 -1.
As with the probit model, there is no simple formula for the MLE of the logit coefficients, so
the log likelihood must be maximized numerically.
Pseudo-R2
The pseudo-R2 compares the value of the likelihood of the estimated model to the value of the
likelihood when none of the X’s are included as regressors. Specifically, the pseudo-R2 for the
probit model is
pseudo - R2 = 1 -
ln1f max
probit 2
ln1f max
Bernoulli 2
,
(11.18)
max
wheref max
probit is the value of the maximized probit likelihood (which includes the X’s) and f Bernoulli
is the value of the maximized Bernoulli likelihood (the probit model excluding all the X’s).
Standard Errors for Predicted Probabilities
For simplicity, consider the case of a single regressor in the probit model. Then the predicted
and
+ bnMLE
x2, where bnMLE
probability at a fixed value of that regressor, x, is pn 1x2 = Φ1bnMLE
0
1
0
bnMLE
are the MLEs of the two probit coefficients. Because this predicted probability depends
1
and bnMLE
, and because those estimators have a sampling distribution,
on the estimators bnMLE
0
1
the predicted probability will also have a sampling distribution.
The variance of the sampling distribution of pn 1x2 is calculated by approximating the
and bnMLE
, by a linear function of
+ bnMLE
x2, a nonlinear function of bnMLE
function Φ1bnMLE
0
1
0
1
bnMLE
and bnMLE
. Specifically, let
0
1
pn 1x2 = Φ1bnMLE
+ bnMLE
x2 ≅ c + a0 1bnMLE
- b0 2 + a1 1bnMLE
- b1 2,
0
1
0
1
(11.19)
where the constant c and factors a0 and a1 depend on x and are obtained from calculus.
[Equation (11.19) is a first-order Taylor series expansion; c = Φ1b0 + b1x2; and a0 and a1 are
and a1 = 0 Φ1b0 + b1x2>0b1 ! bN MLE
.]
the partial derivatives, a0 = 0 Φ1b0 + b1x2>0b0 ! bN MLE
, bN MLE
, bN MLE
0
1
0
1
The variance of pn 1x2 now can be calculated using the approximation in Equation (11.19) and
the expression for the variance of the sum of two random variables in Equation (2.32):
var3pn 1x24 ≅ var3c + a0 1bnMLE
- b0 2 + a1 1bnMLE
- b1 24
0
1
n MLE
= a20var1bnMLE
2 + a21var1bnMLE
2 + 2a0a1cov1bnMLE
,b
2.
0
1
0
1
(11.20)
Using Equation (11.20), the standard error of pn 1x2 can be calculated using estimates of the
variances and covariance of the MLEs.
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CHAPTER 11!!Regression with a Binary Dependent Variable
APPENDIX
11.3 Other Limited Dependent Variable Models
This appendix surveys some models for limited dependent variables, other than binary
variables, found in econometric applications. In most cases, the OLS estimators of the
parameters of limited dependent variable models are inconsistent, and estimation is routinely done using maximum likelihood. There are several advanced references available to
the reader interested in further details; see, for example, Greene (2018), Ruud (2000), and
Wooldridge (2010).
Censored and Truncated Regression Models
Suppose you have cross-sectional data on car purchases by individuals in a given year. Car
buyers have positive expenditures, which can reasonably be treated as continuous random
variables, but nonbuyers spend $0. Thus the distribution of car expenditures is a combination
of a discrete distribution (at 0) and a continuous distribution.
Nobel laureate James Tobin developed a useful model for a dependent variable with a
partly continuous and partly discrete distribution (Tobin, 1958). Tobin suggested modeling the
ith individual in the sample as having a desired level of spending, Y *i , that is related to the
regressors (for example, family size) according to a linear regression model. That is, when there
is a single regressor, the desired level of spending is
Y *i = b0 + b1Xi + u i, i = 1, c, n.
(11.21)
If Y *i (what the consumer wants to spend) exceeds some cutoff, such as the minimum price of
a car, the consumer buys the car and spends Yi = Y *i , which is observed. However, if Y *i is less
than the cutoff, spending of Yi = 0 is observed instead of Y *i .
When Equation (11.21) is estimated using observed expenditures Yi in place of Y *i , the
OLS estimator is inconsistent. Tobin solved this problem by deriving the likelihood function using the additional assumption that ui has a normal distribution, and the resulting
MLE has been used by applied econometricians to analyze many problems in economics.
In Tobin’s honor, Equation (11.21), combined with the assumption of normal errors, is
called the tobit regression model. The tobit model is an example of a censored regression
model, so called because the dependent variable has been “censored” above or below a
certain cutoff.
Sample Selection Models
In the censored regression model, there are data on buyers and nonbuyers, as there would be
if the data were obtained via simple random sampling of the adult population. If, however, the
data are collected from sales tax records, then the data would include only buyers: There would
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Other Limited Dependent Variable Models
383
be no data at all for nonbuyers. Data in which observations are unavailable above or below a
threshold (data for buyers only) are called truncated data. The truncated regression model is a
regression model applied to data in which observations are simply unavailable when the
dependent variable is above or below a certain cutoff.
The truncated regression model is an example of a sample selection model, in which the
selection mechanism (an individual is in the sample by virtue of buying a car) is related to the
value of the dependent variable (expenditure on a car). As discussed in the box “James Heckman and Daniel McFadden, Nobel Laureates” in Section 11.5, one approach to estimation of
sample selection models is to develop two equations, one for Y *i and one for whether Y *i is
observed. The parameters of the model can then be estimated by maximum likelihood, or, in
a stepwise procedure, estimating the selection equation first and then estimating the equation
for Y *i . For additional discussion, see Ruud (2000, Chapter 28), Greene (2018, Chapter 19), or
Wooldridge (2010, Chapter 17).
Count Data
Count data arise when the dependent variable is a counting number—for example, the number of restaurant meals eaten by a consumer in a week. When these numbers are large, the
variable can be treated as approximately continuous, but when they are small, the continuous
approximation is a poor one. The linear regression model, estimated by OLS, can be used for
count data, even if the number of counts is small. Predicted values from the regression are
interpreted as the expected value of the dependent variable, conditional on the regressors. So
when the dependent variable is the number of restaurant meals eaten, a predicted value of
1.7 means, on average, 1.7 restaurant meals per week. As in the binary regression model,
however, OLS does not take advantage of the special structure of count data and can yield
nonsense predictions: for example, - 0.2 restaurant meals per week. Just as probit and logit
eliminate nonsense predictions when the dependent variable is binary, special models do so
for count data. The two most widely used models are the Poisson and negative binomial
regression models.
Ordered Responses
Ordered response data arise when mutually exclusive qualitative categories have a natural
ordering, such as obtaining a high school diploma, obtaining some college education (but not
graduating), or graduating from college. Like count data, ordered response data have a natural
ordering, but unlike count data, they do not have natural numerical values.
Because there are no natural numerical values for ordered response data, OLS is inappropriate. Instead, ordered data are often analyzed using a generalization of probit called the
ordered probit model, in which the probability of each outcome (e.g., a college education),
conditional on the independent variables (such as parents’ income), is modeled using the
cumulative normal distribution.
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CHAPTER 11!!Regression with a Binary Dependent Variable
Discrete Choice Data
A discrete choice or multiple choice variable can take on multiple unordered qualitative values.
One example in economics is the mode of transport chosen by a commuter: She might take
the subway, ride the bus, drive, or make her way under her own power (walk, bicycle). If we
were to analyze these choices, the dependent variable would have four possible outcomes
(subway, bus, car, and human-powered). These outcomes are not ordered in any natural way.
Instead, the outcomes are a choice among distinct qualitative alternatives.
The econometric task is to model the probability of choosing the various options given
various regressors such as individual characteristics (how far the commuter’s house is from
the!subway station) and the characteristics of each option (the price of the subway). As
discussed in the box in Section 11.5, models for analysis of discrete choice data can be developed from principles of utility maximization. Individual choice probabilities can be expressed
in probit or logit form, and those models are called multinomial probit and multinomial logit
regression models.
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CHAPTER
12
Instrumental Variables
Regression
C
hapter 9 discussed several problems, including omitted variables, errors in
variables, and simultaneous causality, that make the error term correlated with
the regressor. Omitted variable bias can be addressed directly by including the
omitted variable in a multiple regression, but this is only feasible if you have data on
the omitted variable. And sometimes, such as when causality runs both from X to Y
and from Y to X so that there is simultaneous causality bias, multiple regression simply
cannot eliminate the bias. If a direct solution to these problems is either infeasible or
unavailable, a new method is required.
Instrumental variables (IV) regression is a general way to obtain a
consistent estimator of the unknown causal coefficients when the regressor, X, is
correlated with the error term, u. To understand how IV regression works, think
of!the variation in X as having two parts: one part that, for whatever reason, is
correlated with u (this is the part that causes the problems) and a second part
that!is uncorrelated with u. If you had information that allowed you to isolate
the!second part, you could focus on those variations in X that are uncorrelated
with u and disregard the variations in X that bias the OLS estimates. This is, in
fact, what IV regression does. The information about the movements in X that
are!uncorrelated with u is gleaned from one or more additional variables, called
instrumental variables or simply instruments. Instrumental variables regression
uses these additional variables as tools or “instruments” to isolate the movements
in X that are uncorrelated with u, which in turn permits consistent estimation of
the regression coefficients.
The first two sections of this chapter describe the mechanics and assumptions
of IV regression: why IV regression works, what is a valid instrument, and how to
implement and to interpret the most common IV regression method, two stage
least squares. The key to successful empirical analysis using instrumental
variables is finding valid instruments, and Section 12.3 takes up the question of
how to assess whether a set of instruments is valid. As an illustration, Section 12.4
uses IV regression to estimate the elasticity of demand for cigarettes. Finally,
Section 12.5 turns to the difficult question of where valid instruments come from
in the first place.
385
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CHAPTER 12!!Instrumental Variables Regression
12.1 The IV Estimator with a Single Regressor
and a Single Instrument
We start with the case of a single regressor, X, which might be correlated with the
error, u. If X and u are correlated, the OLS estimator is inconsistent; that is, it may
not be close to the true value of the causal coefficient even when the sample is very
large [see Equation (6.1)]. As discussed in Section 9.2, this correlation between X and
u can stem from various sources, including omitted variables, errors in variables
(measurement errors in the regressors), and simultaneous causality (when causality
runs “backward” from Y to X as well as “forward” from X to Y). Whatever the source
of the correlation between X and u, if there is a valid instrumental variable, Z, the
effect on Y of a unit change in X can be estimated using the instrumental variables
estimator.
The IV Model and Assumptions
Let b1 be the causal effect of X on Y. The model relating the dependent variable Yi
and regressor Xi, without any control variables, is
Yi = b0 + b1Xi + u i, i = 1, c, n,
(12.1)
where ui is the error term representing omitted factors that determine Yi. If Xi and
ui are correlated, the OLS estimator is inconsistent. Instrumental variables estimation uses an additional, “instrumental” variable Z to isolate that part of X that is
uncorrelated with u.
Endogeneity and exogeneity. Instrumental variables regression has some specialized terminology to distinguish variables that are correlated with the population
error term u from ones that are not. Variables correlated with the error term are
called endogenous variables, while variables uncorrelated with the error term are
called exogenous variables. The historical source of these terms traces to models with
multiple equations, in which an “endogenous” variable is determined within the
model, while an “exogenous” variable is determined outside the model. For example,
Section 9.2 considered the possibility that if low test scores produced decreases in the
student–teacher ratio because of political intervention and increased funding, causality would run both from the student–teacher ratio to test scores and from test scores
to the student–teacher ratio. This was represented mathematically as a system of two
simultaneous equations [Equations (9.3) and (9.4)], one for each causal connection.
As discussed in Section 9.2, because both test scores and the student–teacher ratio
are determined within the model, both are correlated with the population error term
u; that is, in this example, both variables are endogenous. In contrast, an exogenous
variable, which is determined outside the model, is uncorrelated with u.
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12.1 !!The IV Estimator with a Single Regressor and a Single Instrument
387
The two conditions for a valid instrument. A valid instrumental variable (“instrument”) Z must satisfy two conditions, known as the instrument relevance condition
and the instrument exogeneity condition:
1. Instrument relevance: corr1Zi, Xi 2 ≠ 0.
2. Instrument exogeneity: corr1Zi, u i 2 = 0.
If an instrument is relevant, then variation in the instrument is related to variation in Xi. If in addition the instrument is exogenous, then that part of the variation
of Xi captured by the instrumental variable is exogenous. Thus an instrument that is
relevant and exogenous can capture movements in Xi that are exogenous. This
exogenous variation can in turn be used to estimate the population coefficient b1.
The two conditions for a valid instrument are vital for instrumental variables
regression, and we return to them (and their extension to multiple regressors and
multiple instruments) repeatedly throughout this chapter.
The Two Stage Least Squares Estimator
If the instrument Z satisfies the conditions of instrument relevance and exogeneity, the
coefficient b1 can be estimated using an IV estimator called two stage least squares (TSLS).
As the name suggests, the two stage least squares estimator is calculated in two stages. The
first stage decomposes X into two components: a problematic component that may be
correlated with the regression error and another, problem-free component that is uncorrelated with the error. The second stage uses the problem-free component to estimate b1.
The first stage begins with a population regression linking X and Z:
Xi = p0 + p1Zi + vi,
(12.2)
where p0 is the intercept, p1 is the slope, and vi is the error term. This regression provides the needed decomposition of Xi. One component is p0 + p1Zi, the part of Xi
that can be predicted by Zi. Because Zi is exogenous, this component of Xi is uncorrelated with ui, the error term in Equation (12.1). The other component of Xi is vi,
which is the problematic component of Xi that is correlated with ui.
The idea behind TSLS is to use the problem-free component of Xi, p0 + p1Zi,
and to disregard vi. The only complication is that the values of p0 and p1 are unknown,
so p0 + p1Zi cannot be calculated. Accordingly, the first stage of TSLS applies OLS
to Equation (12.2) and uses the predicted value from the OLS regression,
ni = p
n0 + p
n 1Zi, where p
n 0 and p
n 1 are the OLS estimates.
X
n i using OLS. The resulting
The second stage of TSLS is easy: Regress Yi on X
estimators from the second-stage regression are the TSLS estimators, bnTSLS
and bnTSLS
.
0
1
Why Does IV Regression Work?
Two examples provide some insight into why IV regression solves the problem of
correlation between Xi and ui.
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CHAPTER 12!!Instrumental Variables Regression
Who Invented Instrumental Variables Regression?
I
nstrumental variables regression was first pro-
Stylometrics is the subfield of statistics, invented
posed in Philip G. Wright’s 1928 book, The Tariff
by Frederick Mosteller and David Wallace (1963),
on Animal and Vegetable Oils. If you yearn to know
that uses subtle, subconscious differences in writ-
how animal and vegetable oils were produced,
ing styles to identify authorship of disputed texts
transported, and sold in the early 20th century, the
using statistical analysis of grammatical construc-
first 285 pages of the book are for you. Econometri-
tions and word choice. When stylometrics is used to
cians, however, are more interested in Appendix B.
compare Appendix B to texts known to have been
The appendix explains the simultaneous causal-
written independently by Philip and by Sewall, the
ity problem and provides two derivations of “the
results are clear: Philip was the author (see Stock
method of introducing external factors”—what
and Trebbi, 2003).
we now call the instrumental variables estimator.
Does this mean that Philip Wright invented IV
It then uses IV regression to estimate the supply
regression? Not quite. Recently, correspondence
and demand elasticities for butter and flaxseed oil.
between Philip and Sewall has come to light. In a
Philip Wright was an obscure economist who never
series of letters between December 1925 and April
held a major academic position, but his son Sewall
1926, the two worked through the challenge of
went on to become a preeminent population geneti-
simultaneous causality and together invented IV
cist and statistician. Many econometricians assumed
regression. Sewall provided one derivation, which
that Sewall wrote the appendix anonymously, but
is very difficult to follow. In fact, Philip admitted
the mystery of who really wrote the appendix, and
he could not follow his son’s derivation, so Philip
who invented IV regression, remained unsolved—
provided an entirely different, much easier deriva-
until recently.
tion. Philip’s derivation, from a letter of March 4,
Either father or son could have been the author.
1926, is reproduced here. Philip’s derivation is the
Philip Wright (1861–1934) received a master’s
one used today, and we use it in Equations (12.5)
degree in economics from Harvard University in
and (12.6).
1887, and he taught mathematics and economics (as
Between the statistical sleuthing and the letters,
well as literature and physical education) at a small
the mystery is solved: Philip Wright wrote
college in Illinois. In a book review (Wright, 1915),
Appendix B, but the invention of IV was a joint col-
he used a figure like Figures 12.1a and 12.1b to show
laboration between father and son.
how a regression of quantity on price will not, in gen-
In the early 1920s, Sewall Wright (1889–1988) was
researching the statistical analysis of multiple equations with multiple causal variables in the context of
genetics, research that in part led to his assuming a
professorship in 1930 at the University of Chicago.
The first clue to the mystery came from some
statistical detective work using textual data analysis.
M12_STOC1991_04_SE_C12_pp385-431.indd 388
Courtesy of Rosalind W. Harris
a combination of the supply and demand curves.
Courtesy of Rosalind W. Harris
eral, estimate a demand curve but instead estimates
Philip G. Wright
Sewall Wright
22/08/18 1:07 PM
389
12.1 !!The IV Estimator with a Single Regressor and a Single Instrument
Philip Wright’s Derivation of the IV Estimator of the Supply Elasticity in His Letter of March 4, 1926,
to His Son Sewall.
Philip’s notation differs from ours: O is quantity (Y in our notation), P is price (X), S is other determinants of supply (the error
term in the supply equation, u), and A is the external factor
(the instrument Z). All variables are in percentage deviations
with mean 0 (we use logs and an intercept), and e is the supply
elasticity (b). The subscripts denote observation number.
The first line rearranges Y = bX + u to be bX = Y - u. He
multiplies both sides by the instrument and uses instrument
exogeneity (“since A is uncorrelated with S”) to set the term
n
n
g i = 1 AiSi (in our notation, g i = 1 Ziui) to 0 in expectation. He
n
then uses instrument relevance to divide by g i = 1 AiPi (in our
n
notation, g i = 1 ZiXi), thereby obtaining the formula for the IV
estimator, which, in our notation, is Equation (12.4).
Example 1: Philip Wright’s problem. The method of instrumental variables estimation was first published in 1928 in an appendix to a book written by Philip G. Wright
(1928), although the key ideas of IV regression were developed collaboratively with
his son Sewall Wright (see the box “Who Invented Instrumental Variables Regression?”). Philip Wright was concerned with an important economic problem of his
day: how to set an import tariff (a tax on imported goods) on animal and vegetable
oils and fats, such as butter and soy oil. In the 1920s, import tariffs were a major
source of tax revenue for the United States. The key to understanding the economic
effect of a tariff was having quantitative estimates of the demand and supply curves
of the goods. Recall that the supply elasticity is the percentage change in the quantity
supplied arising from a 1% increase in the price and that the demand elasticity is the
percentage change in the quantity demanded arising from a 1% increase in the price.
Philip Wright needed estimates of these elasticities of supply and demand.
To be concrete, consider the problem of estimating the elasticity of demand for
butter. Recall from Key Concept 8.2 that the coefficient in a linear equation relating
ln(Yi) to ln(Xi) has the interpretation of the elasticity of Y with respect to X. In
Wright’s problem, this suggests the demand equation
ln 1Qbutter
2 = b0 + b1 ln1P butter
2 + u i,
i
i
(12.3)
where Qbutter
is the ith observation on the quantity of butter consumed, P butter
is its
i
i
price, and ui represents other factors that affect demand, such as income and consumer tastes. In Equation (12.3), a 1% increase in the price of butter yields a b1
percent change in demand, so b1 is the demand elasticity.
Philip Wright had data on total annual butter consumption and its average
annual price in the United States for 1912 to 1922. It would have been easy to use
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CHAPTER 12!!Instrumental Variables Regression
these data to estimate the demand elasticity by applying OLS to Equation (12.3), but
he had a key insight: Because of the interactions between supply and demand, the
regressor, ln1P butter
2, was likely to be correlated with the error term.
i
To see this, look at Figure 12.1a, which shows the market demand and supply
curves for butter for three different years. The demand and supply curves for the first
period are denoted D1 and S1, and the first period’s equilibrium price and quantity
are determined by their intersection. In year 2, demand increases from D1 to D2 (say,
because of an increase in income), and supply decreases from S1 to S2 (because of an
increase in the cost of producing butter); the equilibrium price and quantity are
determined by the intersection of the new supply and demand curves. In year 3, the
factors affecting demand and supply change again; demand increases again to D3,
supply increases to S3, and a new equilibrium quantity and price are determined.
Figure 12.1b shows the equilibrium quantity and price pairs for these three periods
and for eight subsequent years, where in each year the supply and demand curves are
subject to shifts associated with factors other than price that affect market supply and
demand. This scatterplot is like the one that Wright would have seen when he plotted
his data. As he reasoned, fitting a line to these points by OLS will estimate neither a
demand curve nor a supply curve because the points have been determined by
changes in both demand and supply.
Wright realized that a way to get around this problem was to find some third
variable that shifted supply but did not shift demand. Figure 12.1c shows what happens when such a variable shifts the supply curve but demand remains stable. Now
all of the equilibrium price and quantity pairs lie on a stable demand curve, and the
slope of the demand curve is easily estimated. In the instrumental variable formulation of Wright’s problem, this third variable—the instrumental variable—is correlated with price (it shifts the supply curve, which leads to a change in price) but is
uncorrelated with u (the demand curve remains stable). Wright considered several
potential instrumental variables; one was the weather. For example, below-average
rainfall in a dairy region could impair grazing and thus reduce butter production at
a given price (it would shift the supply curve to the left and increase the equilibrium
price), so dairy-region rainfall satisfies the condition for instrument relevance. But
dairy-region rainfall should not have a direct influence on the demand for butter, so
the correlation between dairy-region rainfall and ui would be 0; that is, dairy-region
rainfall satisfies the condition for instrument exogeneity.
Example 2: Estimating the effect on test scores of class size. Despite controlling for
student and district characteristics, the estimates of the effect on test scores of class size
reported in Part II still might have omitted variable bias resulting from unmeasured
variables such as learning opportunities outside school or the quality of the teachers. If
data on these variables, or on suitable control variables, are unavailable, this omitted
variable bias cannot be addressed by including the variables in the multiple regressions.
Instrumental variables regression provides an alternative approach to this
problem. Consider the following hypothetical example: Some California schools
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12.1 !!The IV Estimator with a Single Regressor and a Single Instrument
391
FIGURE 12.1 Equilibrium Price and Quantity Data
(a) Price and quantity are determined by the intersection of
the supply and demand curves. The equilibrium in the first
period is determined by the intersection of the demand
curve D1 and the supply curve S1. Equilibrium in the second
period is the intersection of D2 and S2, and equilibrium in
the!third period is the intersection of D3 and S3.
Price
Period 2
equilibrium
S2
S1
S3
Period 3
equilibrium
D3
Period 1
equilibrium
D2
D1
Quantity
(a) Demand and supply in three time periods
(b) This scatterplot shows equilibrium price and quantity in
11 different time periods. The demand and supply curves
are hidden. Can you determine the demand and supply
curves from the points on the scatterplot?
Price
Quantity
(b) Equilibrium price and quantity for 11
time periods
(c) When the supply curve shifts from S1 to S2 to S3 but the
demand curve remains at D1, the equilibrium prices and
quantities trace out the demand curve.
Price
S2
S1
S3
D1
Quantity
(c) Equilibrium price and quantity when only
the supply curve shifts
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CHAPTER 12!!Instrumental Variables Regression
are forced to close for repairs because of a summer earthquake. Districts closest
to the epicenter are most severely affected. A district with some closed schools
needs to “double up” its students, temporarily increasing class size. This means
that distance from the epicenter satisfies the condition for instrument relevance
because it is correlated with class size. But if distance to the epicenter is unrelated
to any of the other factors affecting student performance (such as whether the students are still learning English or disruptive effects of the earthquake on student
performance), then it will be exogenous because it is uncorrelated with the error
term. Thus the instrumental variable, distance to the epicenter, could be used to
circumvent omitted variable bias and to estimate the effect of class size on test
scores.
The Sampling Distribution of the TSLS Estimator
The exact distribution of the TSLS estimator in small samples is complicated.
However, like the OLS estimator, its distribution in large samples is simple: The
TSLS estimator is consistent and is normally distributed.
Formula for the TSLS estimator. Although the two stages of TSLS make the
estimator seem complicated, when there is a single X and a single instrument Z, as
we assume in this section, there is a simple formula for the TSLS estimator. Let sZY
be the sample covariance between Z and Y, and let sZX be the sample covariance
between Z and X. As shown in Appendix 12.2, the TSLS estimator with a single
instrument is
sZY
bnTSLS
= s .
1
ZX
(12.4)
That is, the TSLS estimator of b1 is the ratio of the sample covariance between Z and
Y to the sample covariance between Z and X.
Sampling distribution of bnTSLS
when the sample size is large. The formula in Equa1
tion (12.4) can be used to show that bnTSLS
is consistent and, in large samples, normally
1
distributed. The argument is summarized here, with mathematical details given in
Appendix 12.3.
The argument that bnTSLS
is consistent combines the assumptions that Zi is
1
relevant and exogenous with the consistency of sample covariances for population
covariances. To begin, note that because Yi = b0 + b1Xi + u i in Equation (12.1),
cov1Zi, Yi 2 = cov1Zi, b0 + bXi + u i 2 = b1 cov1Zi, Xi 2 + cov1Zi, u i 2, (12.5)
where the second equality follows from the properties of covariances [Equation
(2.34)]. By the instrument exogeneity assumption, cov1Zi, u i 2 = 0, and by the
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12.1 !!The IV Estimator with a Single Regressor and a Single Instrument
393
instrument relevance assumption, cov1Zi, Xi 2 ≠ 0. Thus, if the instrument is valid,
Equation (12.5) implies that
b1 =
cov1Zi, Yi 2
.
cov1Zi, Xi 2
(12.6)
That is, the population coefficient b1 is the ratio of the population covariance between
Z and Y to the population covariance between Z and X.
As discussed in Section 3.7, the sample covariance is a consistent estimator of the
p
p
population covariance; that is, sZY ¡ cov1Zi, Yi 2 and sZX ¡ cov1Zi, Xi 2. It
follows from Equations (12.4) and (12.6) that the TSLS estimator is consistent:
sZY p cov1Zi, Yi 2
bnTSLS
=
¡
= b1.
1
sZX
cov1Zi, Xi 2
(12.7)
The formula in Equation (12.4) also can be used to show that the sampling distribution
of bnTSLS
is normal in large samples. The reason is the same as for every other least
1
squares estimator we have considered: The TSLS estimator is an average of random
variables, and when the sample size is large, the central limit theorem tells us that
averages of random variables are normally distributed. Specifically, the numerator of
n
the expression for bnTSLS
in Equation (12.4) is sZY = n -1 1 g i = 1 1Zi - Z2 1Yi - Y2,
1
an average of 1Zi - Z2 1Yi - Y2. A bit of algebra, sketched out in Appendix 12.3,
shows that because of this averaging, the central limit theorem implies that, in large
samples, bnTSLS
has a sampling distribution that is approximately N1b1, s2bn TSLS 2, where
1
1
s2bn TSLS = 1
1
n
var 3 1Zi - mZ 2u i 4
3cov1Zi, Xi 24 2
.
(12.8)
Statistical inference using the large-sample distribution. The variance s2bn TSLS can be
1
estimated by estimating the variance and covariance terms appearing in Equation
(12.8), and the square root of the estimate of s2bn TSLS is the standard error of the IV
1
estimator. This is done automatically in TSLS regression commands in econometric
software packages. Because bnTSLS
is normally distributed in large samples, hypothesis
1
tests about b1 can be performed by computing the t-statistic, and a 95% large-sample
confidence interval is given by bnTSLS
{ 1.96 SE1bnTSLS
2.
1
1
Application to the Demand for Cigarettes
Philip Wright was interested in the demand elasticity of butter, but today other commodities, such as cigarettes, figure more prominently in public policy debates. One
tool in the quest for reducing illnesses and deaths from smoking—and the!costs, or
externalities, imposed by those illnesses on the rest of society—is to tax cigarettes
so heavily that current smokers cut back and potential new smokers are discouraged
from taking up the habit. But precisely how big a tax hike is needed to make a dent
in cigarette consumption? For example, what would the after-tax!sales price of cigarettes need to be to achieve a 20% reduction in cigarette consumption?
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The answer to this question depends on the elasticity of demand for cigarettes.
If the elasticity is -1, then the 20% target in consumption can be achieved by a 20%
increase in price. If the elasticity is -0.5, then the price must rise 40% to decrease
consumption by 20%. Of course, we do not know the demand elasticity of cigarettes:
We must estimate it from data on prices and sales. But, as with butter, because of the
interactions between supply and demand, the elasticity of demand for cigarettes
cannot be estimated consistently by an OLS regression of log quantity on log price.
We therefore use TSLS to estimate the elasticity of demand for cigarettes using
annual data for the 48 contiguous U.S. states for 1985 through 1995 (the data are
described in Appendix 12.1). For now, all the results are for the cross section of states
in 1995; results using data for earlier years (panel data) are presented in Section 12.4.
The instrumental variable, SalesTaxi, is the portion of the tax on cigarettes arising
from the general sales tax, measured in dollars per pack (in real dollars, deflated by
the Consumer Price Index). Cigarette consumption, Qicigarettes, is the number of packs
of cigarettes sold per capita in the state, and the price, P icigarettes, is the average real
price per pack of cigarettes including all taxes.
Before using TSLS, it is essential to ask whether the two conditions for instrument validity hold. We return to this topic in detail in Section 12.3, where we provide
some statistical tools that help in this assessment. Even with those statistical tools,
judgment plays an important role, so it is useful to think about whether the sales tax
on cigarettes plausibly satisfies the two conditions.
First consider instrument relevance. Because a high sales tax increases the aftertax sales price P icigarettes, the sales tax per pack plausibly satisfies the condition for
instrument relevance.
Next consider instrument exogeneity. For the sales tax to be exogenous, it must be
uncorrelated with the error in the demand equation; that is, the sales tax must affect the
demand for cigarettes only indirectly through the price. This seems plausible: General
sales tax rates vary from state to state, but they do so mainly because different states
choose different mixes of sales, income, property, and other taxes to finance public
undertakings. Those choices about public finance are driven by political considerations,
not by factors related to the demand for cigarettes. We discuss the credibility of this
assumption more in Section 12.4, but for now we keep it as a working hypothesis.
In modern statistical software, the first stage of TSLS is estimated automatically, so you do not need to run this regression yourself to compute the TSLS
estimator. Even so, it is a good idea to look at the first-stage regression. Using data
for the 48 states in 1995, it is
ln1P icigarettes 2 = 4.62 + 0.031SalesTaxi.
10.032 10.0052
(12.9)
As expected, higher sales taxes mean higher after-tax prices. The R2 of this regression
is 47%, so the variation in sales tax on cigarettes explains 47% of the variance of cigarette prices across states.
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395
In the second stage of TSLS, ln1Qicigarettes 2 is regressed on ln1P icigarettes 2 using
OLS. The resulting estimated regression function is
ln1Qicigarettes 2 = 9.72 - 1.08 ln1P icigarettes 2.
(12.10)
This estimated regression function is written using the regressor in the second stage,
the predicted value ln1P icigarettes 2. It is, however, conventional and less cumbersome
simply to report the estimated regression function with ln1P icigarettes 2 rather than
ln1P icigarettes 2. Reported in this notation, the TSLS estimates and heteroskedasticityrobust standard errors are
ln1Qicigarettes 2 = 9.72 - 1.08 ln1P icigarettes 2.
11.532 10.322
(12.11)
The TSLS estimate suggests that the demand for cigarettes is surprisingly elastic in
light of their addictive nature: An increase in the price of 1% reduces consumption
by 1.08%. But, recalling our discussion of instrument exogeneity, perhaps this estimate should not yet be taken too seriously. Even though the elasticity was estimated
using an instrumental variable, there might still be omitted variables that are correlated with the sales tax per pack. A leading candidate is income: States with higher
incomes might depend relatively less on a sales tax and more on an income tax to
finance state government. Moreover, the demand for cigarettes presumably depends
on income. Thus we would like to reestimate our demand equation including income
as a control variable. To do so, however, we must first extend the IV regression model
to include additional regressors.
12.2 The General IV Regression Model
The general IV regression model has four types of variables: the dependent variable, Y; problematic endogenous regressors, like the price of cigarettes, which are
correlated with the error term and which we will label X; additional regressors W,
which are either control variables or included exogenous variables; and instrumental
variables, Z. In general, there can be multiple endogenous regressors (X’s), multiple
additional regressors (W’s), and multiple instrumental variables (Z’s).
For IV regression to be possible, there must be at least as many instrumental variables (Z’s) as endogenous regressors (X’s). In Section 12.1, there was a single endogenous
regressor and a single instrument. Having (at least) one instrument for this single endogenous regressor was essential. Without the instrument, we could not have computed the
instrumental variables estimator: there would be no first-stage regression in TSLS.
The relationship between the number of instruments and the number of endogenous regressors has its own terminology. The regression coefficients are said to be
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CHAPTER 12!!Instrumental Variables Regression
KEY CONCEPT
12.1
The General Instrumental Variables Regression
Model and Terminology
The general IV regression model is
Yi = b0 + b1X1i + g + bkXki + bk + 1W1i + g + bk + rWri + u i,
(12.12)
i = 1, c, n, where
• Yi is the dependent variable;
• b0, b1, c, bk + r are unknown coefficients;
• X1i, c, Xki are k endogenous regressors, which are potentially correlated
with ui;
• W1i, c, Wri are r included exogenous regressors, which are uncorrelated
with ui or are control variables;
• ui is the error term, which represents measurement error and/or omitted
factors; and
• Z1i, c, Zmi are m instrumental variables.
The coefficients are overidentified if there are more instruments than endogenous
regressors 1m 7 k2, they are underidentified if m 6 k, and they are exactly identified if m = k. Estimation of the IV regression model requires exact identification or overidentification.
exactly identified if the number of instruments (m) equals the number of endogenous regressors (k); that is, m = k. The coefficients are overidentified if the number
of instruments exceeds the number of endogenous regressors; that is, m 7 k. They
are underidentified if the number of instruments is less than the number of endogenous regressors; that is, m 6 k. The coefficients must be either exactly identified or
overidentified if they are to be estimated by IV regression.
The general IV regression model and its terminology are summarized in
Key Concept 12.1.
Included exogenous variables and control variables in IV regression. The W variables in Equation (12.12) can be either exogenous variables, in which case
E1u i " Wi 2 = 0, or they can be control variables that need not have a causal interpretation but are included to ensure that the instrument is uncorrelated with the
error term. For example, Section 12.1 raised the possibility that the sales tax might
be correlated with income, which economic theory tells us is a determinant of
cigarette demand. If so, the sales tax would be correlated with the error term in the
cigarette demand equation, ln1Qcigarettes
2 = b0 + b1ln1Picigarettes 2 + u i , and thus
i
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397
would not be an exogenous instrument. Including income in the IV regression, or
including variables that control for income, would remove this source of potential
correlation between the instrument and the error term. In general, if W is an effective control variable in IV regression, then including W makes the instrument
uncorrelated with u, so the TSLS estimator of the coefficient on X is consistent; if
W is correlated with u, however, then the TSLS coefficient on W is subject to omitted variable bias and does not have a causal interpretation. The logic of control
variables in IV regression therefore parallels the logic of control variables in OLS,
discussed in Section 7 .5.
The mathematical condition for W to be an effective control variable in IV
regression is similar to the condition on control variables in OLS discussed in
Section 7 .5. Specifically, including W must ensure that the conditional mean of u
does not depend on Z, so conditional mean independence holds; that is,
E1u i " Zi, Wi 2 = E1u i " Wi 2. For clarity, in the body of this chapter we focus on the case
that W variables are exogenous, so that E1u i " Wi 2 = 0. Appendix 12.6 explains how
the results of this chapter extend to the case that W is a control variable, in which case
the conditional mean 0 condition, E1u i " Wi 2 = 0, is replaced by the conditional mean
independence condition, E1u i " Zi, Wi 2 = E1u i " Wi 2.
TSLS in the General IV Model
TSLS with a single endogenous regressor. When there is a single endogenous regressor X and some additional included exogenous variables, the equation of interest is
Yi = b0 + b1Xi + b2W1i + g + b1 + rWri + u i,
(12.13)
where, as before, Xi might be correlated with the error term, but W1i, c, Wri are not.
The population first-stage regression of TSLS relates X to the exogenous variables—that is, the W’s and the instruments (Z’s):
Xi = p0 + p1Z1i + g + pmZmi + pm + 1W1i + g + pm + rWri + vi, (12.14)
where p0, p1, c, pm + r are unknown regression coefficients and vi is an error term.
Equation (12.14) is sometimes called the reduced form equation for X. It relates
the endogenous variable X to all the available exogenous variables, both those
included in the regression of interest (W) and the instruments (Z).
In the first stage of TSLS, the unknown coefficients in Equation (12.14) are
n 1, c, X
n n.
estimated by OLS, and the predicted values from this regression are X
In the second stage of TSLS, Equation (12.13) is estimated by OLS except that
Xi is replaced by its predicted value from the first stage. That is, Yi is regressed on
n i, W1i, c, Wri using OLS. The resulting estimator of b0, b1, c, b1 + r is the TSLS
X
estimator.
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CHAPTER 12!!Instrumental Variables Regression
KEY CONCEPT
12.2
Two Stage Least Squares
The TSLS estimator in the general IV regression model in Equation (12.12) with
multiple instrumental variables is computed in two stages:
1. First-stage regression(s): Regress X 1i on the instrumental variables
1Z1i, c, Zmi 2 and the included exogenous variables and/or control variables
1W1i, c, Wri 2 using OLS, including an intercept. Compute the predicted valn 1i. Repeat this for all the endogenous
ues from this regression; call these X
n 1i, c, X
n ki.
regressors X2i, c, Xki, thereby computing the predicted values X
2. Second-stage regression: Regress Yi on the predicted values of the endogenous
n 1i, c, X
n ki 2 and the included exogenous variables and/or control
variables 1X
variables 1W1i, c, Wri 2 using OLS, including an intercept. The TSLS estimators bnTSLS
, c, bnTSLS
0
k + r are the estimators from the second-stage regression.
In practice, the two stages are done automatically within TSLS estimation commands in econometric software.
Extension to multiple endogenous regressors. When there are multiple endogenous
regressors X1i, c, Xki, the TSLS algorithm is similar except that each endogenous
regressor requires its own first-stage regression. Each of these first-stage regressions
has the same form as Equation (12.14); that is, the dependent variable is one of the
X’s, and the regressors are all the instruments (Z’s) and all the included exogenous
variables (W’s). Together, these first-stage regressions produce predicted values of
each of the endogenous regressors.
In the second stage of TSLS, Equation (12.12) is estimated by OLS except that
the endogenous regressors (X’s) are replaced by their respective predicted values
n ’s). The resulting estimator of b0, b1, c, bk + r is the TSLS estimator.
(X
In practice, the two stages of TSLS are done automatically within TSLS estimation commands in econometric software. The general TSLS estimator is summarized
in Key Concept 12.2.
Instrument Relevance and Exogeneity
in the General IV Model
The conditions of instrument relevance and exogeneity need to be modified for the
general IV regression model.
When there is one included endogenous variable but multiple instruments, the
condition for instrument relevance is that at least one Z is useful for predicting X
given W. When there are multiple included endogenous variables, this condition is
more complicated because we must rule out perfect multicollinearity in the secondstage population regression. Intuitively, when there are multiple included
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12.2 !!The General IV Regression Model
The Two Conditions for Valid Instruments
A set of m instruments Z1i, c, Zmi must satisfy the following two conditions to
be valid:
399
KEY CONCEPT
12.3
1. Instrument Relevance
n *1i be the predicted value of X1i from the population regres• In general, let X
sion of X1i on the instruments (Z’s) and the included exogenous regressors
(W’s), and let “1” denote the constant regressor that takes on the value 1
n *1i, c, X
n *ki, W1i, c, Wri, 12 are not perfectly
for all observations. Then 1X
multicollinear.
• If there is only one X, then for the previous condition to hold, at least one
Z must have a nonzero coefficient in the population regression of X on the
Z’s and the W’s.
2. Instrument Exogeneity
The instruments are uncorrelated with the error term; that is, corr1Z1i, u i 2 =
0, c, corr1Zmi, u i 2 = 0.
endogenous variables, the instruments must provide enough information about the
exogenous movements in these variables to sort out their separate effects on Y.
The general statement of the instrument exogeneity condition is that each instrument must be uncorrelated with the error term ui. The general conditions for valid
instruments are given in Key Concept 12.3.
The IV Regression Assumptions and Sampling
Distribution of the TSLS Estimator
Under the IV regression assumptions, the TSLS estimator is consistent and has a
sampling distribution that, in large samples, is approximately normal.
The IV regression assumptions. The IV regression assumptions are modifications of
the least squares assumptions for causal inference in the multiple regression model
in Key Concept 6.4.
The first IV regression assumption modifies the conditional mean assumption in
Key Concept 6.4 to apply only to the included exogenous variables. Just like the
second least squares assumption for the multiple regression model, the second IV
regression assumption is that the draws are i.i.d., as they are if the data are collected
by simple random sampling. Similarly, the third IV assumption is that large outliers
are unlikely.
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CHAPTER 12!!Instrumental Variables Regression
KEY CONCEPT
12.4
The IV Regression Assumptions
The variables and errors in the IV regression model in Key Concept 12.1 satisfy
the following:
1. E1u i " W1i, c, Wri 2 = 0;
2. 1X1i, c, Xki, W1i, c, Wri, Z1i, c, Zmi, Yi 2 are i.i.d. draws from their joint
distribution;
3. Large outliers are unlikely: The X’s, W’s, Z’s, and Y have nonzero finite fourth
moments; and
4. The two conditions for a valid instrument in Key Concept 12.3 hold.
The fourth IV regression assumption is that the two conditions for instrument
validity in Key Concept 12.3 hold. The instrument relevance condition in Key Concept 12.3 subsumes the fourth least squares assumption in Key Concepts 6.4 and 6.6
(no perfect multicollinearity) by assuming that the regressors in the second-stage
regression are not perfectly multicollinear. The IV regression assumptions are summarized in Key Concept 12.4.
Sampling distribution of the TSLS estimator. Under the IV regression assumptions,
the TSLS estimator is consistent and normally distributed in large samples. This is
shown in Section 12.1 (and Appendix 12.3) for the special case of a single endogenous regressor, a single instrument, and no included exogenous variables. Conceptually, the reasoning in Section 12.1 carries over to the general case of multiple
instruments and multiple included endogenous variables. The expressions in the general case are complicated, however, and are deferred to Chapter 19.
Inference Using the TSLS Estimator
Because the sampling distribution of the TSLS estimator is normal in large samples,
the general procedures for statistical inference (hypothesis tests and confidence
intervals) in regression models extend to TSLS regression. For example, 95% confidence intervals are constructed as the TSLS estimator {1.96 standard errors. Similarly, joint hypotheses about the population values of the coefficients can be tested
using the F-statistic, as described in Section 7 .2.
Calculation of TSLS standard errors. There are two points to bear in mind about
TSLS standard errors. First, the standard errors reported by OLS estimation of the
second-stage regression are incorrect because they do not recognize that it is the
second stage of a two-stage process. Specifically, the second-stage OLS standard
errors fail to adjust for the second-stage regression using the predicted values of the
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12.2 !!The General IV Regression Model
401
included endogenous variables. Formulas for standard errors that make the necessary adjustment are incorporated into (and automatically used by) TSLS regression
commands in econometric software. Therefore, this issue is not a concern in practice
if you use a specialized TSLS regression command.
Second, as always the error u might be heteroskedastic. It is therefore important
to use heteroskedasticity-robust versions of the standard errors for precisely the
same reason that it is important to use heteroskedasticity-robust standard errors for
the OLS estimators of the multiple regression model.
Application to the Demand for Cigarettes
In Section 12.1, we estimated the elasticity of demand for cigarettes using data on
annual consumption in 48 U.S. states in 1995 using TSLS with a single regressor (the
logarithm of the real price per pack) and a single instrument (the real sales tax per
pack). Income also affects demand, however, so it is part of the error term of the
population regression. As discussed in Section 12.1, if the state sales tax is related to
state income, it is correlated with a variable in the error term of the cigarette demand
equation, which violates the instrument exogeneity condition. If so, the IV estimator
in Section 12.1 is inconsistent. That is, the IV regression suffers from a version of
omitted variable bias. We can solve this problem by including income in the
regression.
We therefore consider an alternative specification in which the logarithm of
income is included in the demand equation. In the terminology of Key Concept 12.1,
the dependent variable Y is the logarithm of consumption, ln1Qicigarettes 2; the endogenous regressor X is the logarithm of the real after-tax price, ln1P icigarettes 2; the
included exogenous variable W is the logarithm of the real per capita state income,
ln(Inci); and the instrument Z is the real sales tax per pack, SalesTaxi. The TSLS estimates and (heteroskedasticity-robust) standard errors are
ln1Qicigarettes 2 = 9.43 - 1.14 ln1P icigarettes 2 + 0.21 ln1Inci 2.
(1.26) (0.37)
(0.31)
(12.15)
This regression uses a single instrument, SalesTaxi, but, in fact, another candidate
instrument is available. In addition to general sales taxes, states levy special taxes that
apply only to cigarettes and other tobacco products. These cigarette-specific taxes
(CigTaxi) constitute a possible second instrumental variable. The cigarette-specific
tax increases the price of cigarettes paid by the consumer, so it arguably meets the
condition for instrument relevance. If it is uncorrelated with the error term in the
state cigarette demand equation, it is an exogenous instrument.
With this additional instrument in hand, we now have two instrumental variables,
the real sales tax per pack and the real state cigarette-specific tax per pack. With two
instruments and a single endogenous regressor, the demand elasticity is overidentified; that is, the number of instruments (SalesTaxi and CigTaxi, so m = 2) exceeds
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CHAPTER 12!!Instrumental Variables Regression
the number of included endogenous variables (P icigarettes, so k = 1). We can estimate
the demand elasticity using TSLS, where the regressors in the first-stage regression
are the included exogenous variable, ln(Inci), and both instruments.
The resulting TSLS estimate of the regression function using the two instruments
SalesTaxi and CigTaxi is
ln1Qicigarettes 2 = 9.89 - 1.28 ln1P icigarettes 2 + 0.28 ln1Inci 2 .
(0.96) (0.25)
(0.25)
(12.16)
Compare Equations (12.15) and (12.16): The standard error of the estimated price
elasticity is smaller by one-third in Equation (12.16) [0.25 in Equation (12.16) versus
0.37 in Equation (12.15)]. The reason the standard error is smaller in Equation (12.16)
is that this estimate uses more information than Equation (12.15): In Equation
(12.15), only one instrument (the sales tax) is used, but in Equation (12.16), two
instruments (the sales tax and the cigarette-specific tax) are used. Using two instruments explains more of the variation in cigarette prices than using just one, and this
is reflected in smaller standard errors on the estimated demand elasticity.
Are these estimates credible? Ultimately, credibility depends on whether the set
of instrumental variables—here, the two taxes—plausibly satisfies the two conditions
for valid instruments. It is therefore vital that we assess whether these instruments
are valid, and it is to this topic that we now turn.
12.3 Checking Instrument Validity
Whether instrumental variables regression is useful in a given application hinges on
whether the instruments are valid: Invalid instruments produce meaningless results.
It therefore is essential to assess whether a given set of instruments is valid in a particular application.
Assumption 1: Instrument Relevance
The role of the instrument relevance condition in IV regression is subtle. One way to
think of instrument relevance is that it plays a role akin to the sample size: The more
relevant are the instruments—that is, the more the variation in X is explained by
the!instruments—the more information is available for use in IV regression. A more
relevant instrument produces a more accurate estimator, just as a larger sample size
produces a more accurate estimator. Moreover, statistical inference using TSLS is
predicated on the TSLS estimator having a normal sampling distribution, but according to the central limit theorem, the normal distribution is a good approximation in
large—but not necessarily small—samples. If having a more relevant instrument is
like having a larger sample size, this suggests, correctly, that the more relevant is the
instrument, the better is the normal approximation to the sampling distribution of
the TSLS estimator and its t-statistic.
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12.3 !!Checking Instrument Validity
403
Instruments that explain little of the variation in X are called weak instruments.
In the cigarette example, the distance of the state from cigarette manufacturing
plants arguably would be a weak instrument: Although a greater distance increases
shipping costs (thus shifting the supply curve in and raising the equilibrium price),
cigarettes are lightweight, so shipping costs are a small component of the price of
cigarettes. Thus the amount of price variation explained by shipping costs, and thus
distance to manufacturing plants, probably is quite small.
This section discusses why weak instruments are a problem, how to check for
weak instruments, and what to do if you have weak instruments. It is assumed
throughout that the instruments are exogenous.
Why weak instruments are a problem. If the instruments are weak, then the normal distribution provides a poor approximation to the sampling distribution of the
TSLS estimator, even if the sample size is large. Thus there is no theoretical justification for the usual methods for performing statistical inference, even in large samples.
In fact, if instruments are weak, then the TSLS estimator can be badly biased in the
direction of the OLS estimator. In addition, 95% confidence intervals constructed
as the TSLS estimator {1.96 standard errors can contain the true value of the coefficient far less than 95% of the time. In short, if instruments are weak, TSLS is no
longer reliable.
To see that there is a problem with the large-sample normal approximation to
the sampling distribution of the TSLS estimator, consider the special case, introduced in Section 12.1, of a single included endogenous variable, a single instrument, and no included exogenous regressor. If the instrument is valid, then bnTSLS
1
is consistent because the sample covariances sZY and sZX are consistent; that is,
p
bnTSLS
= sZY >sZX ¡ cov1Zi, Yi 2>cov1Zi, Xi 2 = b1 [Equation (12.7)]. But now
1
suppose that the instrument is not just weak but in fact is irrelevant, so that
p
cov1Zi, Xi 2 = 0. Then sZX ¡ cov1Zi, Xi 2 = 0, so, taken literally, the denominator on the right-hand side of the limit cov1Zi, Yi 2 >cov1Zi, Xi 2 is 0! Clearly, the
argument that bnTSLS
is consistent breaks down when the instrument relevance
1
condition fails. As shown in Appendix 12.4, this breakdown results in the TSLS
estimator having a nonnormal sampling distribution, even if the sample size is
very large. In fact, when the instrument is irrelevant, the large-sample distribution
of bnTSLS
is not the distribution of a normal random variable but rather the distribu1
tion of a ratio of two normal random variables! As discussed in Appendix 12.4, this
ratio-of-normals distribution is centered at the large-sample value of the OLS
estimator.
While this circumstance of totally irrelevant instruments might not be encountered in practice, it raises a question: How relevant must the instruments be for the
normal distribution to provide a good approximation in practice? The answer to this
question in the general IV model is complicated. Fortunately, however, there is a
simple rule of thumb available for the most common situation in practice, the case of
a single endogenous regressor.
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CHAPTER 12!!Instrumental Variables Regression
KEY CONCEPT
12.5
A Rule of Thumb for Checking for Weak Instruments
The first-stage F-statistic is the F-statistic testing the hypothesis that the coefficients on the instruments Z1i, c, Zmi equal 0 in the first stage of two stage least
squares. When there is a single endogenous regressor, a first-stage F-statistic less
than 10 indicates that the instruments are weak, in which case the TSLS estimator
is biased (even in large samples) and TSLS t-statistics and confidence intervals
are unreliable.
Checking for weak instruments when there is a single endogenous regressor. One
way to check for weak instruments when there is a single endogenous regressor is to
compute the F-statistic testing the hypothesis that the coefficients on the instruments
are all 0 in the first-stage regression of TSLS. This first-stage F-statistic provides a
measure of the information content contained in the instruments: The more information content, the larger the expected value of the F-statistic. One simple rule of
thumb is that you do not need to worry about weak instruments if the first-stage
F-statistic exceeds 10. (Why 10? See Appendix 12.5.) This is summarized in
Key Concept 12.5.
What do I do if I have weak instruments? If you have many instruments, some of
those instruments are probably weaker than others. If you have a small number of
strong instruments and many weak ones, you will be better off discarding the weakest
instruments and using the most relevant subset for your TSLS analysis. Your TSLS
standard errors might increase when you drop weak instruments, but keep in mind
that your original standard errors were not meaningful anyway!
If, however, the coefficients are exactly identified, you cannot discard the weak
instruments. Even if the coefficients are overidentified, you might not have enough
strong instruments to achieve identification, so discarding some weak instruments will
not help. In this case, you have two options. The first option is to find additional, stronger
instruments. This is easier said than done: It requires an intimate knowledge of the problem at hand and can entail redesigning the data set and the nature of the empirical study.
The second option is to proceed with your empirical analysis using the weak instruments, but employing methods other than TSLS. Although this chapter has focused on
TSLS, some other methods for instrumental variable analysis are less sensitive to weak
instruments than TSLS, and some of these methods are discussed in Appendix 12.5.
Assumption 2: Instrument Exogeneity
If the instruments are not exogenous, then TSLS is inconsistent: The TSLS estimator
converges in probability to something other than the causal coefficient. After all, the
idea of instrumental variables regression is that the instrument contains information
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12.3 !!Checking Instrument Validity
405
The First IV Regression
A
fter he and his son Sewall derived the IV esti-
After laborious computations—by hand, of course—
mator (see the box “Who Invented Instrumen-
Philip obtained the IV estimate of the supply elasticity,
tal Variables Regression?”), Philip Wright set out to
-0.88. This elasticity has the wrong sign: It suggests that
see how it worked in practice. In a letter to Sewall of
the supply curve slopes down. In the March 15 letter,
March 15, 1926, Philip wrote out a table (reproduced
Philip called this result “obviously absurd.”
here in part) of annual data on variables relating to
So what went wrong? Although Philip did not
U.S. production of flaxseed from 1903 through 1925.
know it, his IV regression had a first-stage F-statistic
Flaxseed was grown for its oil, also called linseed oil,
of 1.75, far less than the rule-of-thumb cutoff of 10.
which was used in oil-based paint for buildings. Philip
As explained in the text and in Appendix 12.4, when
wanted to estimate the elasticity of supply. To get a per-
the instrument is irrelevant, its distribution centers
cent–percent relationship, he first transformed the data
on the OLS estimate, which in Wright’s data is -0.66.
to be percentage deviations from a long-term trend.
This first IV regression had a very weak instrument,
Philip then needed to make a key decision: What
and the result was biased toward OLS.
instrument should he use? He chose building per-
But Philip persevered. For estimating the demand
mits on the East Coast. He reasoned that if there
elasticity, he had as an instrument rainfall in the
were more new buildings, there would be more
Upper Midwest, where flaxseed was grown. More
demand for oil-based paint and thus for flaxseed, so
rain makes for a better harvest, so rainfall is plau-
the instrument would be relevant. He further rea-
sibly relevant; because rainfall in the Midwest does
soned that fluctuations in building permits on the
not affect the demand for oil paint, it is plausibly
East Coast were largely driven by broader economic
exogenous. Rainfall, it turns out, has a first-stage F of
conditions that had nothing to do with disturbances
12.8 and yields an IV estimate of the demand elastic-
to flaxseed supply in a given year, so that building
ity of -0.48. This estimate indicates that the demand
permits would be exogenous. Said differently, fluc-
curve slopes down (as it should) and that demand is
tuations in building permits on the East Coast were
inelastic, which is consistent with there being no good
a determinant of demand but not of supply.
substitute for linseed oil for paints during this period.
The First Five Observations of the First IV Regression Data Set, from Philip Wright’s Letter to Sewall
Wright of March 15, 1926.
The first two data columns are the real price and quantity (“output”) of flaxseed. The “B” variables— acreage
planted, yield, rainfall in the Upper Midwest, and the ratio of flaxseed yield that year to spring wheat yield the
previous year—shift supply but not demand, so they are potential instruments for the demand elasticity. The “A”
variable—building permits on the East Coast—shifts demand but not supply, so it is a potential instrument for the
supply elasticity.
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CHAPTER 12!!Instrumental Variables Regression
about variation in Xi that is unrelated to the error term ui. If, in fact, the instrument
is not exogenous, it cannot pinpoint this exogenous variation in Xi, and it stands to
reason that IV regression fails to provide a consistent estimator. The math behind
this argument is summarized in Appendix 12.4.
Can you statistically test the assumption that the instruments are exogenous? Yes
and no. On the one hand, it is not possible to test the hypothesis that the instruments
are exogenous when the coefficients are exactly identified. On the other hand, if the
coefficients are overidentified, it is possible to test the overidentifying restrictions—
that is, to test the hypothesis that the “extra” instruments are exogenous under the
maintained assumption that there are enough valid instruments to identify the coefficients of interest.
First consider the case that the coefficients are exactly identified, so you have as
many instruments as endogenous regressors. Then it is impossible to develop a statistical test of the hypothesis that the instruments are, in fact, exogenous. That is,
empirical evidence cannot be brought to bear on the question of whether these
instruments satisfy the exogeneity restriction. In this case, the only way to assess
whether the instruments are exogenous is to draw on expert opinion and your personal knowledge of the empirical problem at hand. For example, Philip Wright’s
knowledge of agricultural supply and demand led him to suggest that below-average
rainfall would plausibly shift the supply curve for fats and oils but would not directly
shift the demand curve.
Assessing whether the instruments are exogenous necessarily requires making
an expert judgment based on personal knowledge of the application. If, however,
there are more instruments than endogenous regressors, then there is a statistical
tool that can be helpful in this process: the so-called test of overidentifying
restrictions.
The overidentifying restrictions test. Suppose you have a single endogenous regressor and two instruments. Then you could compute two different TSLS estimators: one
using the first instrument and the other using the second. These two estimators will
not be the same because of sampling variation, but if both instruments are exogenous, then they will tend to be close to each other. But what if these two instruments
produce very different estimates? You might sensibly conclude that there is something wrong with one or the other of the instruments or with both. That is, it would
be reasonable to conclude that one or the other or both of the instruments are not
exogenous.
The test of overidentifying restrictions implicitly makes this comparison. We say
implicitly because the test is carried out without actually computing all of the different
possible IV estimates. Here is the idea. Exogeneity of the instruments means that they
are uncorrelated with ui. This suggests that the instruments should be approximately
uncorrelated with un TSLS
, where un TSLS
= Yi - 1bnTSLS
+ bnTSLS
X1i + g + bnTSLS
i
i
0
1
k + r Wri 2
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12.3 !!Checking Instrument Validity
The Overidentifying Restrictions Test (The J-Statistic)
Let un TSLS
be the residuals from TSLS estimation of Equation (12.12). Use OLS to
i
estimate the regression coefficients in
407
KEY CONCEPT
12.6
un TSLS
= d0 + d1Z1i + g + dmZmi + dm + 1W1i + g + dm + rWri + ei, (12.17)
i
where ei is the regression error term. Let F denote the homoskedasticity-only
F-statistic testing the hypothesis that d1 = g = dm = 0. The overidentifying
restrictions test statistic is J = mF. Under the null hypothesis that all the instruments are exogenous, if ei is homoskedastic, in large samples J is distributed x2m - k,
where m - k is the degree of overidentification—that is, the number of instruments minus the number of endogenous regressors.
is the residual from the estimated TSLS regression using all the instruments (approximately rather than exactly because of sampling variation). (Note that these residuals
are constructed using the true X’s rather than their first-stage predicted values.)
Accordingly, if the instruments are, in fact, exogenous, then the coefficients on the
instruments in a regression of un TSLS
on the instruments and the included exogenous
i
variables should all be 0, and this hypothesis can be tested.
This method for computing the overidentifying restrictions test is summarized in
Key Concept 12.6. This statistic is computed using the homoskedasticity-only F-statistic.
The test statistic is commonly called the J-statistic and is computed as J = mF.
In large samples, if the instruments are not weak and the errors are homoskedastic, then, under the null hypothesis that the instruments are exogenous, the J-statistic
has a chi-squared distribution with m " k degrees of freedom 1x2m - k 2. It is important
to remember that even though the number of restrictions being tested is m, the
degrees of freedom of the asymptotic distribution of the J-statistic is m - k. The
reason is that it is possible to test only the overidentifying restrictions, of which there
are m - k. The modification of the J-statistic for heteroskedastic errors is given in
Section 19.7.
The easiest way to see that you cannot test the exogeneity of the regressors when
the coefficients are exactly identified 1m = k2 is to consider the case of a single
included endogenous variable 1k = 12. If there are two instruments, then you can
compute two TSLS estimators, one for each instrument, and you can compare them
to see if they are close. But if you have only one instrument, then you can compute
only one TSLS estimator, and you have nothing to which to compare it. In fact, if the
coefficients are exactly identified, so that m = k, then the overidentifying test statistic J is exactly 0.
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CHAPTER 12!!Instrumental Variables Regression
12.4 Application to the Demand for Cigarettes1
Our attempt to estimate the elasticity of demand for cigarettes left off with the TSLS
estimates summarized in Equation (12.16), in which income was an included exogenous variable and there were two instruments, the general sales tax and the cigarettespecific tax. We can now undertake a more thorough evaluation of these instruments.
As in Section 12.1, it makes sense that the two instruments are relevant because
taxes are a big part of the after-tax price of cigarettes, and shortly we will look at this
empirically. First, however, we focus on the difficult question of whether the two tax
variables are plausibly exogenous.
The first step in assessing whether an instrument is exogenous is to think through
the arguments for why it may or may not be. This requires thinking about which factors account for the error term in the cigarette demand equation and whether these
factors are plausibly related to the instruments.
Why do some states have higher per capita cigarette consumption than others?
One reason might be variation in incomes across states, but state income is included in
Equation (12.16), so this is not part of the error term. Another reason is that there are
historical factors influencing demand. For example, states that grow tobacco have
higher rates of smoking than most other states. Could this factor be related to taxes?
Quite possibly: If tobacco farming and cigarette production are important industries in
a state, then these industries could exert influence to keep cigarette-specific taxes low.
This suggests that an omitted factor in cigarette demand—whether the state grows
tobacco and produces cigarettes—could be correlated with cigarette-specific taxes.
One solution to this possible correlation between the error term and the instrument would be to include information on the size of the tobacco and cigarette industry in the state; this is the approach we took when we included income as a regressor
in the demand equation. But because we have panel data on cigarette consumption,
a different approach is available that does not require this information. As discussed
in Chapter 10, panel data make it possible to eliminate the influence of variables that
vary across entities (states) but do not change over time, such as the historical circumstances that lead to a large tobacco and cigarette industry in a state. Two methods
for doing this were given in Chapter 10: constructing data on changes in the variables
between two different time periods and using fixed effects regression. To keep the
analysis here as simple as possible, we adopt the former approach and perform
regressions of the type described in Section 10.2, based on the changes in the variables between two different years.
The time span between the two different years influences how the estimated elasticities are to be interpreted. Because cigarettes are addictive, changes in price will
take some time to alter behavior. At first, an increase in the price of cigarettes might
have little effect on demand. Over time, however, the price increase might contribute
1
This section assumes knowledge of the material in Sections 10.1 and 10.2 on panel data with T = 2 time
periods.
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12.4 !!Application to the Demand for Cigarettes
409
The Externalities of Smoking
S
moking imposes costs that are not fully borne
death, the net present value of the per-pack external-
by the smoker; that is, it generates externalities.
ities (the value of the net costs per pack, discounted
One economic justification for taxing cigarettes
to the present) depends on the discount rate.
therefore is to “internalize” these externalities.
The studies do not agree on a specific dollar value
In theory, the tax on a pack of cigarettes should
of the net externalities. Some suggest that the net
equal the dollar value of the externalities created by
externalities, properly discounted, are quite small,
smoking that pack. But what, precisely, are the exter-
less than current taxes. In fact, the most extreme
nalities of smoking, measured in dollars per pack?
estimates suggest that the net externalities are posi-
Several studies have used econometric methods
tive, so smoking should be subsidized! Other studies,
to estimate the externalities of smoking. The negative
which incorporate costs that are probably important
externalities—costs—borne by others include medi-
but difficult to quantify (such as caring for babies
cal costs paid by the government to care for ill smok-
who are unhealthy because their mothers smoke),
ers, health care costs of nonsmokers associated with
suggest that externalities might be $1 per pack, pos-
secondhand smoke, and fires caused by cigarettes.
sibly even more. But all the studies agree that, by
But, from a purely economic point of view, smok-
tending to die in late middle age, smokers pay far
ing also has positive externalities, or benefits. The
more in taxes than they ever get back in their brief
biggest economic benefit of smoking is that smok-
retirement.1
ers tend to pay much more in Social Security (public
pension) taxes than they ever get back. There are also
large savings in nursing home expenditures on the
very old—smokers tend not to live that long. Because
the negative externalities of smoking occur while the
smoker is alive but the positive ones accrue after
1
An early calculation of the externalities of smoking was
reported by Willard G. Manning et al. (1989). A calculation suggesting that health care costs would go up if everyone stopped smoking is presented in Barendregt, Bonneux,
and van der Maas (1997). Other studies of the externalities
of smoking are reviewed by Chaloupka and Warner (2000).
to some smokers’ desire to quit, and, importantly, it could discourage nonsmokers
from taking up the habit. Thus the response of demand to a price increase could be
small in the short run but large in the long run. Said differently, for an addictive product like cigarettes, demand might be inelastic in the short run—that is, it might have a
short-run elasticity near 0—but it might be more elastic in the long run.
In this analysis, we focus on estimating the long-run price elasticity. We do
this by considering quantity and price changes that occur over 10-year periods.
Specifically, in the regressions considered here, the 10-year change in log
cigarettes
cigarettes
quantity, ln1Qi,1995
2 - ln1Qi,1985
2, is regressed against the 10-year change in
cigarettes
cigarettes
log! price, ln1P i,1995 2 - ln1P i,1985 2, and the 10-year change in log income,
ln1Inci,1995 2 - ln1Inci,1985 2. Two instruments are used: the change in the sales tax
over 10 years, SalesTaxi,1995 - SalesTaxi,1985, and the change in the cigarette-specific
tax over 10 years, CigTaxi,1995 - CigTaxi,1985.
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CHAPTER 12!!Instrumental Variables Regression
TABLE 12.1
Two Stage Least Squares Estimates of the Demand for Cigarettes Using
Panel Data for 48 U.S. States
Dependent variable: ln1Q i,cigarettes
2 − ln1Q i,cigarettes
2
1995
1985
Regressor
(1)
(2)
(3)
- ln1P cigarettes
i,1985 2
- 0.94
(0.21)
[ - 1.36, - 0.52]
- 1.34
(0.23)
[ - 1.80, - 0.88]
- 1.20
(0.20)
[ - 1.60, - 0.81]
ln1Inci,1995 2 - ln1Inci,1985 2
0.53
(0.34)
[ - 0.16, 1.21]
0.43
(0.30)
[ - 0.16, 1.02]
0.46
(0.31)
[ - 0.16, 1.09]
Intercept
- 0.12
(0.07)
- 0.02
(0.07)
- 0.05
(0.06)
Instrumental variable(s)
Sales tax
Cigarette-specific tax
Both sales tax and
cigarette-specific tax
First-stage F-statistic
33.7
107 .2
88.6
Overidentifying restrictions
J-test and p-value
—
—
ln1P cigarettes
i,1995 2
4.93
(0.026)
These regressions were estimated using data for 48 U.S. states (48 observations on the 10-year differences). The data are
described in Appendix 12.1. The J-test of overidentifying restrictions is described in Key Concept 12.6 (its p-value is given
in parentheses), and the first-stage F-statistic is described in Key Concept 12.5. Heteroskedasticity-robust standard errors
are given in parentheses beneath coefficients, and 95% confidence intervals are given in brackets.
The results are presented in Table 12.1. As usual, each column in the table presents the results of a different regression. All regressions have the same regressors, and
all coefficients are estimated using TSLS; the only difference among the three regressions is the set of instruments used. In column (1), the only instrument is the sales
tax; in column (2), the only instrument is the cigarette-specific tax; and in column (3),
both taxes are used as instruments.
In IV regression, the reliability of the coefficient estimates hinges on the validity
of the instruments, so the first things to look at in Table 12.1 are the diagnostic statistics assessing the validity of the instruments.
First, are the instruments relevant? We need to look at the first-stage F-statistics.
The first-stage regression in column (1) is
cigarettes
cigarettes
ln1P i,1995
2 - ln1P i,1985
2 = 0.53 - 0.223ln1Inci,1995 2 - ln1Inci,1985 24
(0.03) (0.22)
+ 0.02551SalesTaxi,1995 - SalesTaxi,1985 2. (12.18)
(0.0044)
Because there is only one instrument in this regression, the first-stage F-statistic
is the square of the t-statistic testing that the coefficient on the instrumental variable,
SalesTaxi,1995 - SalesTaxi,1985, is 0; this is F = t 2 = 10.0255 > 0.00442 2 = 33.7. For the
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12.4 !!Application to the Demand for Cigarettes
411
regressions in columns (2) and (3), the first-stage F-statistics are 107.2 and 88.6, so in
all three cases the first-stage F-statistics exceed 10. We conclude that the instruments
are not weak, so we can rely on the standard methods for statistical inference (hypothesis tests and confidence intervals) using the TSLS coefficients and standard errors.
Second, are the instruments exogenous? Because the regressions in columns (1)
and (2) each have a single instrument and a single included endogenous regressor,
the coefficients in those regressions are exactly identified. Thus we cannot deploy the
J-test in either of those regressions. The regression in column (3), however, is overidentified because there are two instruments and a single included endogenous
regressor, so there is one 1m - k = 2 - 1 = 12 overidentifying restriction. The
J-statistic is 4.93; this has a x21 distribution, so the 5% critical value is 3.84 (Appendix
Table 3 ) and the null hypothesis that both the instruments are exogenous is rejected
at the 5% significance level (this deduction also can be made directly from the
p-value of 0.026, reported in the table).
The reason the J-statistic rejects the null hypothesis that both instruments are
exogenous is that the two instruments produce rather different estimated coefficients.
When the only instrument is the sales tax [column (1)], the estimated price elasticity
is -0.94, but when the only instrument is the cigarette-specific tax, the estimated price
elasticity is -1.34. Recall the basic idea of the J-statistic: If both instruments are exogenous, then the two TSLS estimators using the individual instruments are consistent
and differ from each other only because of random sampling variation. If, however,
one of the instruments is exogenous and one is not, then the estimator based on the
endogenous instrument is inconsistent, which is detected by the J-statistic. In this
application, the difference between the two estimated price elasticities is sufficiently
large that it is unlikely to be the result of pure sampling variation, so the J-statistic
rejects the null hypothesis that both the instruments are exogenous.
The J-statistic rejection means that the regression in column (3) is based on
invalid instruments (the instrument exogeneity condition fails). What does this imply
about the estimates in columns (1) and (2)? The J-statistic rejection says that at least
one of the instruments is endogenous, so there are three logical possibilities: The sales
tax is exogenous but the cigarette-specific tax is not, in which case the column (1)
regression is reliable; the cigarette-specific tax is exogenous but the sales tax is not,
so the column (2) regression is reliable; or neither tax is exogenous, so neither regression is reliable. The statistical evidence cannot tell us which possibility is correct, so
we must use our judgment.
We think that the case for the exogeneity of the general sales tax is stronger than
that for the cigarette-specific tax because the political process can link changes in the
cigarette-specific tax to changes in the cigarette market and smoking policy. For
example, if smoking decreases in a state because it falls out of fashion, there will be
fewer smokers and a weakened lobby against cigarette-specific tax increases, which
in turn could lead to higher cigarette-specific taxes. Thus changes in tastes (which are
part of u) could be correlated with changes in cigarette-specific taxes (the instrument). This suggests discounting the IV estimates that use the cigarette-only tax as
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CHAPTER 12!!Instrumental Variables Regression
an instrument and adopting the price elasticity estimated using the general sales tax
as an instrument, -0.94.
The estimate of -0.94 indicates that cigarette consumption is somewhat elastic:
An increase in price of 1% leads to a decrease in consumption of 0.94%. This may
seem surprising for an addictive product like cigarettes. But remember that this elasticity is computed using changes over a 10-year period, so it is a long-run elasticity.
This estimate suggests that increased taxes can make a substantial dent in cigarette
consumption, at least in the long run.
When the elasticity is estimated using 5-year changes from 1985 to 1990 rather
than the 10-year changes reported in Table 12.1, the elasticity (estimated with the
general sales tax as the instrument) is -0.79; for changes from 1990 to 1995, the
elasticity is -0.68. These estimates suggest that demand is less elastic over horizons
of 5 years than over 10 years. This finding of greater price elasticity at longer horizons
is consistent with the large body of research on cigarette demand. Demand elasticity
estimates in that literature typically fall in the range -0.3 to -0.5, but these are
mainly short-run elasticities; some studies suggest that the long-run elasticity could
be perhaps twice the short-run elasticity.2
12.5 Where Do Valid Instruments Come From?
In practice, the most difficult aspect of IV estimation is finding instruments that are
both relevant and exogenous. There are two main approaches, which reflect two different perspectives on econometric and statistical modeling.
The first approach is to use economic theory to suggest instruments. For example, Philip Wright’s understanding of the economics of agricultural markets led him
to look for an instrument that shifted the supply curve but not the demand curve; this
in turn led him to consider weather conditions in agricultural regions. One area
where this approach has been particularly successful is the field of financial economics. Some economic models of investor behavior involve statements about how investors forecast, which then imply sets of variables that are uncorrelated with the error
term. Those models sometimes are nonlinear in the data and in the parameters, in
which case the IV estimators discussed in this chapter cannot be used. An extension
of IV methods to nonlinear models, called generalized method of moments estimation, is used instead. Economic theories are, however, abstractions that often do not
take into account the nuances and details necessary for analyzing a particular data
set. Thus this approach does not always work.
The second approach to constructing instruments is to look for some exogenous
source of variation in X arising from what is, in effect, a random phenomenon that
2
A sobering economic study by Adda and Cornaglia (2006) suggests that smokers compensate for higher
taxes by smoking more intensively, thus extracting more nicotine per cigarette. If you are interested in
learning more about the economics of smoking, see Chaloupka and Warner (2000), Gruber (2001), and
Carpenter and Cook (2008).
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12.5 !!Where Do Valid Instruments Come From?
413
induces shifts in the endogenous regressor. For example, in our hypothetical example
in Section 12.1, earthquake damage increased average class size in some school districts, and this variation in class size was unrelated to potential omitted variables that
affect student achievement. This approach typically requires knowledge of the problem being studied and careful attention to the details of the data, and it is best
explained through examples.
Three Examples
We now turn to three empirical applications of IV regression that illustrate how different researchers used their expert knowledge of their empirical problem to find
instrumental variables.
Does putting criminals in jail reduce crime? This is a question only an economist
would ask. After all, a criminal cannot commit a crime outside jail while in prison,
and that some criminals are caught and jailed serves to deter others. But the magnitude of the combined effect—the change in the crime rate associated with a 1%
increase in the prison population—is an empirical question.
One strategy for estimating this effect is to regress crime rates (crimes per
100,000 members of the general population) against incarceration rates (prisoners
per 100,000 members of the general population), using annual data at a suitable level
of jurisdiction (for example, U.S. states). This regression could include some control
variables measuring economic conditions (crime increases when general economic
conditions worsen), demographics (youths commit more crimes than the elderly),
and so forth. There is, however, a serious potential for simultaneous causality bias
that undermines such an analysis: If the crime rate goes up and the police do their
job, there will be more prisoners. On the one hand, increased incarceration reduces
the crime rate; on the other hand, an increased crime rate increases incarceration. As
in the butter example in Figure 12.1, because of this simultaneous causality, an OLS
regression of the crime rate on the incarceration rate will estimate some complicated
combination of these two effects. This problem cannot be solved by finding better
control variables.
This simultaneous causality bias, however, can be eliminated by finding a suitable instrumental variable and using TSLS. The instrument must be correlated with
the incarceration rate (it must be relevant), but it must also be uncorrelated with the
error term in the crime rate equation of interest (it must be exogenous). That is, it
must affect the incarceration rate but be unrelated to any of the unobserved factors
that determine the crime rate.
Where does one find something that affects incarceration but has no direct effect
on the crime rate? One place is exogenous variation in the capacity of existing prisons. Because it takes time to build a prison, short-term capacity restrictions can force
states to release prisoners prematurely or otherwise reduce incarceration rates. Using
this reasoning, Levitt (1996) suggested that lawsuits aimed at reducing prison
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CHAPTER 12!!Instrumental Variables Regression
overcrowding could serve as an instrumental variable, and he implemented this idea
using panel data for the U.S. states from 1972 to 1993.
Are variables measuring overcrowding litigation valid instruments? Although
Levitt did not report first-stage F-statistics, the prison overcrowding litigation slowed
the growth of prisoner incarcerations in his data, suggesting that this instrument is
relevant. To the extent that overcrowding litigation is induced by prison conditions
but not by the crime rate or its determinants, this instrument is exogenous. Because
Levitt breaks down overcrowding legislation into several types and thus has several
instruments, he is able to test the overidentifying restrictions and fails to reject them
using the J-statistic, which bolsters the case that his instruments are valid.
Using these instruments and TSLS, Levitt estimated the effect on the crime rate
of incarceration to be substantial. This estimated effect was three times larger than
the effect estimated using OLS, suggesting that OLS suffered from large simultaneous causality bias.
Does cutting class sizes increase test scores? As we saw in the empirical analysis of
Part II, schools with small classes tend to be wealthier, and their students have access
to enhanced learning opportunities both in and out of the classroom. In Part II, we
used multiple regression to tackle the threat of omitted variables bias by controlling
for various measures of student affluence, ability to speak English, and so forth. Still,
a skeptic could wonder whether we did enough: If we left out something important,
our estimates of the class size effect would still be biased.
This potential omitted variables bias could be addressed by including the right
control variables, but if these data are unavailable (some, like outside learning opportunities, are hard to measure), then an alternative approach is to use IV regression.
This regression requires an instrumental variable correlated with class size (relevance) but uncorrelated with the omitted determinants of test performance that
make up the error term, such as parental interest in learning, learning opportunities
outside the classroom, quality of the teachers and school facilities, and so forth
(exogeneity).
Where does one look for an instrument that induces random, exogenous variation in class size, but is unrelated to the other determinants of test performance?
Hoxby (2000) suggested biology. Because of random fluctuations in timings of births,
the size of the incoming kindergarten class varies from one year to the next. Although
the actual number of children entering kindergarten might be endogenous (recent
news about the school might influence whether parents send a child to a private
school), she argued that the potential number of children entering kindergarten—the
number of 4-year-olds in the district—is mainly a matter of random fluctuations in
the birth dates of children.
Is potential enrollment a valid instrument? Whether it is exogenous depends on
whether it is correlated with unobserved determinants of test performance. Surely
biological fluctuations in potential enrollment are exogenous, but potential enrollment also fluctuates because parents with young children choose to move into an
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12.5 !!Where Do Valid Instruments Come From?
415
improving school district and out of one in trouble. If so, an increase in potential
enrollment could be correlated with unobserved factors such as the quality of school
management, rendering this instrument invalid. Hoxby addressed this problem by
reasoning that growth or decline in the potential student pool for this reason would
occur smoothly over several years, whereas random fluctuations in birth dates would
produce short-term “spikes” in potential enrollment. Thus she used as her instrument
not potential enrollment, but the deviation of potential enrollment from its longterm trend. These deviations satisfy the criterion for instrument relevance (the firststage F-statistics all exceed 100). She makes a good case that this instrument is
exogenous, but, as in all IV analysis, the credibility of this assumption is ultimately a
matter of judgment.
Hoxby implemented this strategy using detailed panel data on elementary
schools in Connecticut in the 1980s and 1990s. The panel data set permitted her to
include school fixed effects, which, in addition to the instrumental variables strategy,
attack the problem of omitted variables bias at the school level. Her TSLS estimates
suggested that the effect on test scores of class size is small; most of her estimates
were statistically insignificantly different from 0.
Does aggressive treatment of heart attacks prolong lives? Aggressive treatments
for victims of heart attacks (technically, acute myocardial infarctions, or AMIs) hold
the potential for saving lives. Before a new medical procedure—in this example,
cardiac catheterization3—is approved for general use, it goes through clinical trials,
a series of randomized controlled experiments designed to measure its effects and
side effects. But strong performance in a clinical trial is one thing; actual performance
in the real world is another.
A natural starting point for estimating the real-world effect of cardiac catheterization is to compare patients who received the treatment to those who did not.
This leads to regressing the length of survival of the patient against the binary
treatment variable (whether the patient received cardiac catheterization) and
other control variables that affect mortality (age, weight, other measured health
conditions, and so forth). The population coefficient on the indicator variable is the
increment to the patient’s life expectancy provided by the treatment. Unfortunately, the OLS estimator is subject to bias: Cardiac catheterization does not “just
happen” to a patient randomly; rather, it is performed because the doctor and
patient decide that it might be effective. If their decision is based in part on unobserved factors relevant to health outcomes not in the data set, the treatment decision will be correlated with the regression error term. If the healthiest patients are
the ones who receive the treatment, the OLS estimator will be biased (treatment is
correlated with an omitted variable), and the treatment will appear more effective
than it really is.
3
Cardiac catheterization is a procedure in which a catheter, or tube, is inserted into a blood vessel and
guided all the way to the heart to obtain information about the heart and coronary arteries.
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CHAPTER 12!!Instrumental Variables Regression
This potential bias can be eliminated by IV regression using a valid instrumental
variable. The instrument must be correlated with treatment (must be relevant) but
must be uncorrelated with the omitted health factors that affect survival (must be
exogenous).
Where does one look for something that affects treatment but does not affect
the health outcome other than through its effect on treatment? McClellan, McNeil,
and Newhouse (1994) suggested geography. Most hospitals in their data set did not
offer cardiac catheterization, so many patients were closer to “regular” hospitals
that did not offer this treatment than to cardiac catheterization hospitals. McClellan,
McNeil, and Newhouse therefore used as an instrumental variable the difference
between the distance from the AMI patient’s home to the nearest cardiac catheterization hospital and the distance to the nearest hospital of any sort; this distance is
0 if the nearest hospital is a cardiac catheterization hospital, and otherwise it is
positive. If this relative distance affects the probability of receiving this treatment,
then it is relevant. If it is distributed randomly across AMI victims, then it is
exogenous.
Is relative distance to the nearest cardiac catheterization hospital a valid instrument? McClellan, McNeil, and Newhouse do not report first-stage F-statistics, but
they do provide other empirical evidence that it is not weak. Is this distance measure
exogenous? They make two arguments. First, they draw on their medical expertise
and knowledge of the health care system to argue that distance to a hospital is plausibly uncorrelated with any of the unobservable variables that determine AMI outcomes. Second, they have data on some of the additional variables that affect AMI
outcomes, such as the weight of the patient, and in their sample, distance is uncorrelated with these observable determinants of survival; this, they argue, makes it more
credible that distance is uncorrelated with the unobservable determinants in the
error term as well.
Using 205,021 observations on Americans aged at least 64 who had an AMI in
1987, McClellan, McNeil, and Newhouse reached a striking conclusion: Their TSLS
estimates suggest that cardiac catheterization has a small, possibly 0, effect on health
outcomes; that is, cardiac catheterization does not substantially prolong life. In contrast, the OLS estimates suggest a large positive effect. They interpret this difference
as evidence of bias in the OLS estimates.
McClellan, McNeil, and Newhouse’s IV method has an interesting interpretation.
The OLS analysis used actual treatment as the regressor, but because actual treatment is itself the outcome of a decision by patient and doctor, they argue that the
actual treatment is correlated with the error term. Instead, TSLS uses predicted treatment, where the variation in predicted treatment arises because of variation in the
instrumental variable: Patients closer to a cardiac catheterization hospital are more
likely to receive this treatment.
This interpretation has two implications. First, the IV regression actually estimates the effect of the treatment not on a “typical” randomly selected patient but
rather on patients for whom distance is an important consideration in the treatment
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12.6 !!Conclusion
417
decision. The effect on those patients might differ from the effect on a typical patient,
which provides one explanation of the greater estimated effectiveness of the treatment in clinical trials than in McClellan, McNeil, and Newhouse’s IV study. Second,
it suggests a general strategy for finding instruments in this type of setting: Find an
instrument that affects the probability of treatment, but does so for reasons that are
unrelated to the outcome except through their effect on the likelihood of treatment.
Both these implications have applicability to experimental and “quasi-experimental”
studies, the topic of Chapter 13.
12.6 Conclusion
From the humble start of estimating how much less butter people will buy if its price
rises, IV methods have evolved into a general approach for estimating regressions
when one or more variables are correlated with the error term. Instrumental variables regression uses the instruments to isolate variation in the endogenous regressors that is uncorrelated with the error in the regression of interest; this is the first
stage of two stage least squares. This in turn permits estimation of the effect of interest in the second stage of two stage least squares.
Successful IV regression requires valid instruments—that is, instruments that are
both relevant (not weak) and exogenous. If the instruments are weak, then the TSLS
estimator can be biased, even in large samples, and statistical inferences based on
TSLS t-statistics and confidence intervals can be misleading. Fortunately, when there
is a single endogenous regressor, it is possible to check for weak instruments simply
by checking the first-stage F-statistic.
If the instruments are not exogenous—that is, if one or more instruments are
correlated with the error term—the TSLS estimator is inconsistent. If there are more
instruments than endogenous regressors, instrument exogeneity can be examined by
using the J-statistic to test the overidentifying restrictions. However, the core assumption—that there are at least as many exogenous instruments as there are endogenous
regressors—cannot be tested. It is therefore incumbent on both the empirical analyst
and the critical reader to use their own understanding of the empirical application to
evaluate whether this assumption is reasonable.
The interpretation of IV regression as a way to exploit known exogenous variation in the endogenous regressor can be used to guide the search for potential instrumental variables in a particular application. This interpretation underlies much of the
empirical analysis in the area that goes under the broad heading of program evaluation, in which experiments or quasi-experiments are used to estimate the effect of
programs, policies, or other interventions on some outcome measure. A variety of
additional issues arises in those applications—for example, the interpretation of IV
results when, as in the cardiac catheterization example, different “patients” might
have different responses to the same “treatment.” These and other aspects of empirical program evaluation are taken up in Chapter 13.
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CHAPTER 12!!Instrumental Variables Regression
Summary
1. Instrumental variables regression is a way to estimate causal coefficients when
one or more regressors are correlated with the error term.
2. Endogenous variables are correlated with the error term in the equation of
interest; exogenous variables are uncorrelated with this error term.
3. For an instrument to be valid, it must be (1) correlated with the included
endogenous variable and (2) exogenous.
4. IV regression requires at least as many instruments as included endogenous
variables.
5. The TSLS estimator has two stages. First, the included endogenous variables
are regressed against the included exogenous variables and the instruments.
Second, the dependent variable is regressed against the included exogenous
variables and the predicted values of the included endogenous variables from
the first-stage regression(s).
6. Weak instruments (instruments that are nearly uncorrelated with the included
endogenous variables) make the TSLS estimator biased and TSLS confidence
intervals and hypothesis tests unreliable.
7. If an instrument is not exogenous, the TSLS estimator is inconsistent.
Key Terms
instrumental variables (IV)
regression!(385)
instrumental variable (instrument)
(385)
endogenous variable (386)
exogenous variable (386)
instrument relevance condition (387)
instrument exogeneity condition (387)
two stage least squares (387)
included exogenous variables (395)
exactly identified (396)
overidentified (396)
underidentified (396)
reduced form (397)
first-stage regression (398)
second-stage regression (398)
weak instruments (403)
first-stage F-statistic (404)
test of overidentifying restrictions
(406)
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Exercises
419
Review the Concepts
12.1
In the demand curve model of Equation (12.3), is ln 1P butter
2 positively or
i
negatively correlated with the error, ui? If b1 is estimated by OLS, would you
expect the estimated value to be larger or smaller than the true value of b1?
Explain.
12.2
In the study of cigarette demand in this chapter, suppose we used as an instrument the number of trees per capita in the state. Is this instrument relevant?
Is it exogenous? Is it a valid instrument?
12.3
In his study of the effect of incarceration on crime rates, suppose Levitt had
used the number of lawyers per capita as an instrument. Would this instrument be relevant? Would it be exogenous? Would it be a valid instrument?
12.4
In their study of the effectiveness of cardiac catheterization, McClellan,
McNeil, and Newhouse (1994) used as an instrument the difference in distances to cardiac catheterization and regular hospitals. How could you determine whether this instrument is relevant? How could you determine whether
this instrument is exogenous?
Exercises
12.1
This question refers to the panel data IV regressions summarized in Table 12.1.
a. Suppose the federal government is considering a new tax on cigarettes
that is estimated to increase the retail price by $0.50 per pack. If the
current price per pack is $7.50, use the IV regression in column (1) to
predict the change in demand. Construct a 95% confidence interval
for!the change in demand.
b. Suppose the United States enters a recession and income falls by 2%.
Use the IV regression in column (1) to predict the change in demand.
c. Suppose the recession lasts less than one year. Do you think that the IV
regression in column (1) will provide a reliable answer to the question in
(b)? Why or why not?
d. Suppose the F-statistic in column (1) were 3.7 instead of 33.7. Would the
IV regression provide a reliable answer to the question posed in (a)?
Why or why not?
12.2
Consider the regression model with a single regressor: Yi = b0 + b1Xi + u i.
Suppose the least squares assumptions in Key Concept 4.3 are satisfied.
a. Show that Xi is a valid instrument. That is, show that Key Concept 12.3 is
satisfied with Zi = Xi.
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CHAPTER 12!!Instrumental Variables Regression
b. Show that the IV regression assumptions in Key Concept 12.4 are
satisfied with this choice of Zi.
c. Show that the IV estimator constructed using Zi = Xi is identical to the
OLS estimator.
12.3
A classmate is interested in estimating the variance of the error term in Equation (12.1).
a. Suppose she uses the estimator from the second-stage regression of
n i 2 2, where X
n i is the fitted
n 2a = n -1 2 g ni = 1 1Yi - bnTSLS
TSLS: s
- bnTSLS
X
0
1
value from the first-stage regression. Is this estimator consistent? (For
the purposes of this question, suppose that the sample is very large and
the TSLS estimators are essentially identical to b0 and b1.)
n 2b = 1 g n 1Yi - bnTSLS
b. Is s
- bnTSLS
Xi 2 2 consistent?
0
1
n-2
i=1
12.4
Consider TSLS estimation with a single included endogenous variable and a
single instrument. Then the predicted value from the first-stage regression is
ni = p
n0 + p
n 1Zi. Use the definition of the sample variance and covariance to
X
n 1sZY and s2n = p
n 21s2Z. Use this result to fill in the steps of the
show that sXn Y = p
X
derivation of Equation (12.4) in Appendix 12.2.
12.5
Consider the IV regression model
Yi = b0 + b1Xi + b2Wi + u i,
where Xi is correlated with ui and Zi is an instrument. Suppose that the first
three assumptions in Key Concept 12.4 are satisfied. Which IV assumption is
not satisfied when
a. Zi is independent of (Yi, Xi, Wi)?
b. Zi = Wi ?
c. Wi = 1 for all i?
d. Zi = Xi ?
12.6
In an IV regression model with one regressor, Xi, and one instrument, Zi,
the regression of Xi onto Zi has R2 = 0.05 and n = 100. Is Zi a strong instrument? [Hint: See Equation (7.14).] Would your answer change if R2 = 0.05
and n = 500 ?
12.7
In an IV regression model with one regressor, Xi, and two instruments, Z1i and
Z2i, the value of the J-statistic is J = 18.2.
a. Does this suggest that E1u i " Z1i, Z2i 2 ≠ 0 ? Explain.
12.8
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b. Does this suggest that E1u i " Z1i 2 ≠ 0 ? Explain.
Consider a product market with a supply function Qsi = b0 + b1Pi + usi,
a demand function Qdi = g0 + udi, and a market equilibrium condition
12/07/18 10:12 AM
Empirical Exercises
421
Qsi = Qdi, where usi and usi are mutually independent i.i.d. random variables,
both with a mean of 0.
a. Show that Pi and usi are correlated.
b. Show that the OLS estimator of b1 is inconsistent.
c. How would you estimate b0, b1, and g0 ?
12.9
A researcher is interested in the effect of military service on human capital.
He collects data from a random sample of 4000 workers aged 40 and runs the
OLS regression Yi = b0 + b1Xi + u i, where Yi is a worker’s annual earnings
and Xi is a binary variable that is equal to 1 if the person served in the military
and is equal to 0 otherwise.
a. Explain why the OLS estimates are likely to be unreliable. (Hint: Which
variables are omitted from the regression? Are they correlated with
military service?)
b. During the Vietnam War, there was a draft in which priority for the
draft was determined by a national lottery. (The days of the year were
randomly reordered 1 through 365. Those with birth dates ordered first
were drafted before those with birth dates ordered second, and so forth.)
Explain how the lottery might be used as an instrument to estimate the
effect of military service on earnings. (For more about this issue, see
Joshua D. Angrist (1990).)
12.10 Consider the IV regression model Yi = b0 + b1Xi + b2Wi + u i, where Zi is an
instrument. Suppose data on Wi are not available and the model is estimated
omitting Wi from the regression.
a. Suppose Zi and Wi are uncorrelated. Is the IV estimator consistent?
b. Suppose Zi and Wi are correlated. Is the IV estimator consistent?
Empirical Exercises
E12.1
How does fertility affect labor supply? That is, how much does a woman’s
labor supply fall when she has an additional child? In this exercise, you will
estimate this effect using data for married women from the 1980 U.S. Census.4
The data are available on the text website, http://www.pearsonhighered.com/
stock_watson, in the file Fertility and described in the file Fertility_Description.
The data set contains information on married women aged 21–35 with two or
more children.
4
These data were provided by Professor William Evans of the University of Maryland and were used in
his paper with Joshua Angrist, “Children and Their Parents’ Labor Supply: Evidence from Exogenous
Variation in Family Size,” American Economic Review, 1998, 88(3): 450–477.
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CHAPTER 12!!Instrumental Variables Regression
a. Regress weeksworked on the indicator variable morekids, using OLS. On
average, do women with more than two children work less than women
with two children? How much less?
b. Explain why the OLS regression estimated in (a) is inappropriate for
estimating the causal effect of fertility (morekids) on labor supply
(weeksworked).
c. The data set contains the variable samesex, which is equal to 1 if the first
two children are of the same sex (boy–boy or girl–girl) and equal to 0
otherwise. Are couples whose first two children are of the same sex more
likely to have a third child? Is the effect large? Is it statistically significant?
d. Explain why samesex is a valid instrument for the IV regression of
weeksworked on morekids.
e. Is samesex a weak instrument?
f. Estimate the IV regression of weeksworked on morekids, using samesex
as an instrument. How large is the fertility effect on labor supply?
g. Do the results change when you include the variables agem1, black,
hispan, and othrace in the labor supply regression (treating these variable as exogenous)? Explain why or why not.
E12.2
Does viewing a violent movie lead to violent behavior? If so, the incidence of
violent crimes, such as assaults, should rise following the release of a violent movie
that attracts many viewers. Alternatively, movie viewing may substitute for other
activities (such as alcohol consumption) that lead to violent behavior, so that
assaults should fall when more viewers are attracted to the cinema. On the text
website, http://www.pearsonhighered.com/stock_watson, you will find the data
file Movies, which contains data on the number of assaults and movie attendance
for 516 weekends from 1995 through 2004.5 A detailed description is given in
Movies_Description, available on the website. The data set includes weekend U.S.
attendance for strongly violent movies (such as Hannibal), mildly violent movies
(such as Spider-Man), and nonviolent movies (such as Finding Nemo). The data set
also includes a count of the number of assaults for the same weekend in a subset
of counties in the United States. Finally, the data set includes indicators for year,
month, whether the weekend is a holiday, and various measures of the weather.
a.
i. Regress the logarithm of the number of assaults [ln_assaults =
ln(assaults)] on the year and month indicators. Is there evidence of
seasonality in assaults? That is, do there tend to be more assaults in
some months than others? Explain.
ii. Regress total movie attendance (attend = attend_v + attend_m +
attend_n) on the year and month indicators. Is there evidence of
seasonality in movie attendance? Explain.
5
These are aggregated versions of data provided by Gordon Dahl of University of California–San Diego
and Stefano DellaVigna of University of California–Berkeley and were used in their paper “Does Movie
Violence Increase Violent Crime?” Quarterly Journal of Economics, 2009, 124(2): 677–734.
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Empirical Exercises
423
b. Regress ln_assaults on attend_v, attend_m, attend_n, the year and month
indicators, and the weather and holiday control variables available in the
data set.
i. Based on the regression, does viewing a strongly violent movie
increase or decrease assaults? By how much? Is the estimated effect
statistically significant?
ii. Does attendance at strongly violent movies affect assaults
differently than attendance at moderately violent movies?
Differently than attendance at nonviolent movies?
iii. A strongly violent blockbuster movie is released, and the weekend’s
attendance at strongly violent movies increases by 6 million; meanwhile, attendance falls by 2 million for moderately violent movies
and by 1 million for nonviolent movies. What is the predicted effect
on assaults? Construct a 95% confidence interval for the change
in assaults. [Hint: Review Section 7.3 and material surrounding
Equations (8.7) and (8.8).]
c. It is difficult to control for all the variables that affect assaults and that
might be correlated with movie attendance. For example, the effect of
the weather on assaults and movie attendance is only crudely approximated by the weather variables in the data set. However, the data set
does include a set of instruments—pr_attend_v, pr_attend_m, and pr_
attend_n—that are correlated with attendance but are (arguably) uncorrelated with weekend-specific factors (such as the weather) that affect
both assaults and movie attendance. These instruments use historical
attendance patterns, not information on a particular weekend, to predict a
film’s attendance in a given weekend. For example, if a film’s attendance
is high in the second week of its release, then this can be used to predict
that its attendance was also high in the first week of its release. (The
details of the construction of these instruments are available in the Dahl
and DellaVigna paper referenced in footnote 5.) Run the regression from
(b) (including year, month, holiday, and weather controls) but now using
pr_attend_v, pr_attend_m, and pr_attend_n as instruments for attend_v,
attend_m, and attend_n. Use this IV regression to answer (b)(i)–(b)(iii).
d. The intuition underlying the instruments in (c) is that attendance in
a given week is correlated with attendance in surrounding weeks. For
each movie category, the data set includes attendance in surrounding
weeks. Run the regression using the instruments attend_v_f, attend_m_f,
attend_n_f, attend_v_b, attend_m_b, and attend_n_b instead of the instruments used in (c). Use this IV regression to answer (b)(i)–(b)(iii).
e. There are nine instruments listed in (c) and (d), but only three are
needed for identification. Carry out the test for overidentification summarized in Key Concept 12.6. What do you conclude about the validity
of the instruments?
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CHAPTER 12!!Instrumental Variables Regression
f. Based on your analysis, what do you conclude about the effect of violent
movies on (short-run) violent behavior?
E12.3
(This requires Appendix 12.5) On the text website, http://www.pearson
highered.com/stock_watson, you will find the data set WeakInstrument,
which contains 200 observations on (Yi, Xi, Zi) for the instrumental regression
Yi = b0 + b1Xi + u i.
a. Construct bnTSLS
, its standard error, and the usual 95% confidence
1
interval for b1.
b. Compute the F-statistic for the regression of Xi on Zi. Is there evidence
of a “weak instrument” problem?
c. Compute a 95% confidence interval for b1, using the Anderson–Rubin
procedure. (To implement the procedure, assume that -5 … b1 … 5.)
d. Comment on the differences in the confidence intervals in (a) and (c).
Which is more reliable?
APPENDIX
12.1 The Cigarette Consumption Panel Data Set
The data set consists of annual data for the 48 contiguous U.S. states from 1985 to 1995. Quantity consumed is measured by annual per capita cigarette sales in packs per fiscal year, as
derived from state tax collection data. The price is the real (that is, inflation-adjusted) average
retail cigarette price per pack during the fiscal year, including taxes. Income is real per capita
income. The general sales tax is the average tax, in cents per pack, due to the broad-based state
sales tax applied to all consumption goods. The cigarette-specific tax is the tax applied to cigarettes only. All prices, income, and taxes used in the regressions in this chapter are deflated by
the Consumer Price Index and thus are in constant (real) dollars. We are grateful to Professor
Jonathan Gruber of MIT for providing us with these data.
APPENDIX
12.2 Derivation of the Formula for the TSLS
Estimator in Equation (12.4)
The first stage of TSLS is to regress Xi on the instrument Zi by OLS and then compute the
n i; the second stage is to regress Yi on X
n i by OLS. Accordingly, the forOLS predicted value X
n i, is the formula for
mula for the TSLS estimator, expressed in terms of the predicted value X
TSLS
n i replacing Xi. That is, bn1
= s n >s2 , where s2
the OLS estimator in Key Concept 4.2, with X
XY
n i and s n is the sample covariance between Yi and X
n i.
is the sample variance of X
XY
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Xn
Xn
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Large-Sample Distribution of the TSLS Estimator
425
n i is the predicted value of Xi from the first-stage regression, X
ni = p
n0 + p
n 1Zi,
Because X
n 1sZY and s2n = p
n 21s2Z
the definitions of sample variances and covariances imply that sXn Y = p
X
n 1s2Z).
= s n >s2 = sZY >(p
(Exercise 12.4). Thus, the TSLS estimator can be written as bnTSLS
1
XY
Xn
n 1 is the OLS slope coefficient from the first stage of TSLS, so p
n 1 = sZX > s2Z. SubstituFinally, p
n 1 into the formula bnTSLS
n 1s2Z) yields the formula for the TSLS
= sZY >(p
tion of this formula for p
1
estimator in Equation (12.4).
APPENDIX
12.3 Large-Sample Distribution of the TSLS Estimator
This appendix studies the large-sample distribution of the TSLS estimator in the case considered in Section 12.1—that is, with a single instrument, a single included endogenous variable,
and no included exogenous variables.
To start, we derive a formula for the TSLS estimator in terms of the errors; this formula
forms the basis for the remaining discussion, similar to the expression for the OLS estimator
in Equation (4.28) in Appendix 4.3.
From Equation (12.1), Yi - Y = b1 1Xi - X2 + 1u i - u2. Accordingly, the sample cova-
riance between Z and Y can be expressed as
n
1
1Zi - Z21Yi - Y2
n - 1 ia
=1
n
1
=
1Z - Z23b1 1Xi - X2 + 1u i - u24
a
n - 1 i=1 i
n
1
= b1sZX +
1Z - Z21u i - u2
a
n - 1 i=1 i
n
1
= b1sZX +
1Zi - Z2u i,
a
n - 1 i=1
sZY =
(12.19)
where sZX = 31 > 1n - 124 g i = 1 1Zi - Z21Xi - X2 and where the final equality follows
n
because g i = 1 1Zi - Z2 = 0. Substituting the definition of sZX and the final expression in
and multiplying the numerator and denominator
Equation (12.19) into the definition of bnTSLS
1
n
by 1n - 12 > n yields
bnTSLS
= b1 +
1
1 n
1Zi - Z 2u i
n ia
=1
1 n
1Zi - Z 21Xi - X 2
n ia
=1
.
(12.20)
Large-Sample Distribution of bnTSLS
When the IV
1
Regression Assumptions in Key Concept 12.4 Hold
Equation (12.20) for the TSLS estimator is similar to Equation (4.28) in Appendix 4.3 for the
OLS estimator except that Z rather than X appears in the numerator and that the
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CHAPTER 12!!Instrumental Variables Regression
denominator is the covariance between Z and X rather than the variance of X. Because of
these similarities and because Z is exogenous, the argument in Appendix 4.3 that the OLS
.
estimator is normally distributed in large samples extends to bnTSLS
1
q = 1 n1 2 g ni= 1 qi, where qi = 1Zi - mZ 2u i. Because the instrument is exogenous, E1qi 2 = 0.
Specifically, when the sample is large, Z ≅ mZ, so the numerator is approximately
By the IV regression assumptions in Key Concept 12.4, q i is i.i.d. with variance
s2q = var31Zi - mZ 2u i 4. It follows that var 1q2 = sq2 = s2q >n, and, by the central limit theo-
rem, q > sq is, in large samples, distributed N(0, 1).
Because the sample covariance is consistent for the population covariance,
p
sZX ¡ cov1Zi, Xi 2, which, because the instrument is relevant, is nonzero. Thus, by Equation
≅ b1 + q > cov1Zi, Xi 2, so in large samples bnTSLS
is approximately distributed
(12.20), bnTSLS
1
1
N1b1,s2bn TSLS 2 , where s2bn TSLS = sq2 > 3cov1Zi, Xi 24 2 = 11>n2var31Zi - mZ 2u i 4 > 3cov1Zi, Xi 24 2,
1
1
which is the expression given in Equation (12.8).
APPENDIX
12.4 Large-Sample Distribution of the TSLS
Estimator When the Instrument Is Not Valid
This appendix considers the large-sample distribution of the TSLS estimator in the setup of
Section 12.1 (one X, one Z) when one or the other of the conditions for instrument validity
fails. If the instrument relevance condition fails, the large-sample distribution of the TSLS
estimator is not normal; in fact, its distribution is that of a ratio of two normal random variables. If the instrument exogeneity condition fails, the TSLS estimator is inconsistent.
Large-Sample Distribution of bnTSLS
When the
1
Instrument Is Weak
First consider the case that the instrument is irrelevant, so that cov(Zi, Xi) = 0. Then the argument in Appendix 12.3 entails division by 0. To avoid this problem, we need to take a closer
look at the behavior of the term in the denominator of Equation (12.20) when the population
covariance is 0.
We start by rewriting Equation (12.20). Because of the consistency of the sample average,
Equation (12.20) is approximately 1 n1 2 g i = 1 1Zi - mZ 21Xi - mX 2 = 1 n1 2 g i = 1 ri = r, where
in large samples Z is close to mZ, and X is close to mX . Thus the term in the denominator of
n
n
ri = 1Zi - mZ 21Xi - mX 2. Let s2r = var31Zi - mZ 21Xi - mX 24, let s2r = s2r >n, and let q, sq2,
and sq2 be as defined in Appendix 12.3. Then Equation (12.20) implies that, in large samples,
bnTSLS
≅ b1 +
1
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q > sq
sq
sq q > s q
q
= b1 + a b a
b = b1 + a b a
b.
sr
sr
r
r > sr
r > sr
(12.21)
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Instrumental Variables Analysis with Weak Instruments
427
If the instrument is irrelevant, then E1ri 2 = cov1Zi, Xi 2 = 0. Thus r is the sample average of
the random variables ri, i = 1, c, n, which are i.i.d. (by the second least squares assumption),
have variance s2r = var31Zi - mZ 21Xi - mX 24 (which is finite by the third IV regression
assumption), and have a mean of 0 (because the instruments are irrelevant). It follows that the
central limit theorem applies to r; specifically, r > s r is approximately distributed N(0, 1). There-
fore, the final expression of Equation (12.21) implies that, in large samples, the distribution of
bnTSLS
- b1 is the distribution of aS, where a = sq > sr and S is the ratio of two random vari1
ables, each of which has a standard normal distribution (these two standard normal random
variables are correlated).
In other words, when the instrument is irrelevant, the central limit theorem applies to the
denominator as well as the numerator of the TSLS estimator, so in large samples the distribution of the TSLS estimator is the distribution of the ratio of two normal random variables.
Because Xi and ui are correlated, these normal random variables are correlated, and the largesample distribution of the TSLS estimator when the instrument is irrelevant is complicated. In
fact, the large-sample distribution of the TSLS estimator with irrelevant instruments is centered on the probability limit of the OLS estimator. Thus when the instrument is irrelevant,
TSLS does not eliminate the bias in OLS and, moreover, has a nonnormal distribution, even
in large samples.
A weak instrument represents an intermediate case between an irrelevant instrument and
the normal distribution derived in Appendix 12.3. When the instrument is weak but not irrelevant, the distribution of the TSLS estimator continues to be nonnormal, so the general lesson
here about the extreme case of an irrelevant instrument carries over to weak instruments.
Large-Sample Distribution of bnTSLS
1
When the Instrument Is Endogenous
The numerator in the final expression in Equation (12.20) converges in probability to
cov(Zi, ui). If the instrument is exogenous, this is 0, and the TSLS estimator is consistent
(assuming that the instrument is not weak). If, however, the instrument is not exogenous, then,
p
¡ b1 + cov1Zi, u i 2 / cov1Zi, Xi 2 ≠ b1. That is, if the
if the instrument is not weak, bnTSLS
1
instrument is not exogenous, the TSLS estimator is inconsistent.
APPENDIX
12.5 Instrumental Variables Analysis with Weak
Instruments
This appendix discusses some methods for instrumental variables analysis in the presence of
potentially weak instruments. The appendix focuses on the case of a single included endogenous regressor [Equations (12.13) and (12.14)].
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CHAPTER 12!!Instrumental Variables Regression
Testing for Weak Instruments
The rule of thumb in Key Concept 12.5 is that a first-stage F-statistic less than 10 indicates that the
instruments are weak. One motivation for this rule of thumb arises from an approximate expresdenote the probability limit of the OLS estimator
sion for the bias of the TSLS estimator. Let bOLS
1
- b1 denote the asymptotic bias of the OLS estimator (if the regressor is endogb1, and let bOLS
1
p
n
≠ b1). It is possible to show that, when there are many instruments, the
enous, then b1 ¡ bOLS
1
2 - b1 ≈ 1bOLS
- b1 2 > 3E1F2 - 14,
bias of the TSLS estimator is approximately E1bnTSLS
1
1
where E(F) is the expectation of the first-stage F-statistic. If E1F2 = 10, then the bias of TSLS
relative to the bias of OLS is approximately 1/9, or just over 10%, which is small enough to be
acceptable in many applications. Replacing E1F2 7 10 with F 7 10 yields the rule of thumb
in Key Concept 12.5.
The motivation in the previous paragraph involved an approximate formula for the bias
of the TSLS estimator when there are many instruments. In most applications, however, the
number of instruments, m, is small. Stock and Yogo (2005) provide a formal test for weak
instruments that avoids the approximation that m is large. In the Stock–Yogo test, the null
hypothesis is that the instruments are weak, and the alternative hypothesis is that the instruments are strong, where strong instruments are defined to be instruments for which the bias
of the TSLS estimator is at most 10% of the bias of the OLS estimator. The test entails comparing the first-stage F-statistic (for technical reasons, the homoskedasticity-only version) to a
critical value that depends on the number of instruments. As it happens, for a test with a 5%
significance level, this critical value ranges between 9.08 and 11.52, so the rule of thumb of
comparing F to 10 is a good approximation to the Stock–Yogo!test.
Hypothesis Tests and Confidence Sets for b
If the instruments are weak, the TSLS estimator is biased and has a nonnormal distribution.
Thus the TSLS t-test of b1 = b1,0 is unreliable, as is the TSLS confidence interval for b1. There
are, however, other tests of b1 = b1,0, along with confidence intervals based on those tests, that
are valid whether instruments are strong, weak, or even irrelevant. When there is a single
endogenous regressor, the preferred test is Moreira’s (2003) conditional likelihood ratio
(CLR) test. An older test, which works for any number of endogenous regressors, is based on
the Anderson–Rubin (1949) statistic. Because the Anderson–Rubin statistic is conceptually
less complicated, we describe it first.
The Anderson–Rubin test of b1 = b1,0 proceeds in two steps. In the first step, compute a
new variable, Y *i = Yi - b1,0Xi. In the second step, regress Y *i against the included exogenous
regressors (W’s) and the instruments (Z’s). The Anderson–Rubin statistic is the F-statistic
testing the hypothesis that the coefficients on the Z’s are all 0. Under the null hypothesis that
b1 = b1,0, if the instruments satisfy the exogeneity condition (condition 2 in Key Concept
12.3), they will be uncorrelated with the error term in this regression, and the null hypothesis
will be rejected in 5% of all samples.
As discussed in Sections 3.3 and 7.4, a confidence set can be constructed as the set of
values of the parameters that are not rejected by a hypothesis test. Accordingly, the set of
values of b1 that are not rejected by a 5% Anderson–Rubin test constitutes a 95% confidence
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TSLS with Control Variables
429
set for b1. When the Anderson–Rubin F-statistic is computed using the homoskedasticity-only
formula, the Anderson–Rubin confidence set can be constructed by solving a quadratic equation (see Empirical Exercise 12.3). The logic behind the Anderson–Rubin statistic never
assumes instrument relevance, and the Anderson–Rubin confidence set will have a coverage
probability of 95% in large samples, whether the instruments are strong, weak, or even
irrelevant.
The CLR statistic also tests the hypothesis that b1 = b1,0. Likelihood ratio statistics compare the value of the likelihood (see Appendix 11.2) under the null hypothesis to its value
under the alternative and reject it if the likelihood under the alternative is sufficiently greater
than under the null. Familiar test statistics in this text, such as the homoskedasticity-only
F-statistic in multiple regression, can be derived as likelihood ratio statistics under the assumption of homoskedastic normally distributed errors. Unlike any of the other tests discussed in
this text, however, the critical value of the CLR test depends on the data—specifically, on a
statistic that measures the strength of the instruments. By using the right critical value, the
CLR test is valid whether instruments are strong, weak, or irrelevant. CLR confidence intervals can be computed as the set of values of b1 that are not rejected by the CLR test.
The CLR test is equivalent to the TSLS t-test when instruments are strong and has very
good power when instruments are weak. With suitable software, the CLR test is easy to use.
The disadvantage of the CLR test is that it does not generalize readily to more than one
endogenous regressor. In that case, the Anderson–Rubin test (and confidence set) is recommended; however, when instruments are strong (so TSLS is valid) and the coefficients are
overidentified, the Anderson–Rubin test is inefficient in the sense that it is less powerful than
the TSLS t-test.
Estimation of b
If the instruments are irrelevant, then without further restrictions it is not possible to obtain
an unbiased estimator of b1, even in large samples. With weak instruments, CLR or Anderson–
Rubin confidence intervals for the coefficients are preferable to point estimation.
The problems of estimation, testing, and confidence intervals in IV regression with weak
instruments constitute an area of ongoing research. To learn more about this topic, visit the
website for this text.
APPENDIX
12.6 TSLS with Control Variables
In Key Concept 12.4, the W variables are assumed to be exogenous. This appendix considers
the case in which W is not exogenous but instead is a control variable included to make Z
exogenous. The logic of control variables in TSLS parallels the logic in OLS: If a control variable effectively controls for an omitted factor, then the instrument is uncorrelated with the
error term. Because the control variable is correlated with the error term, the coefficient on a
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CHAPTER 12!!Instrumental Variables Regression
control variable does not have a causal interpretation. The mathematics of control variables
in TSLS also parallels the mathematics of control variables in OLS and entails relaxing the
assumption that the error has conditional mean 0 given Z and W to be that the conditional
mean of the error does not depend on Z. This appendix draws on Appendix 6.5 (OLS with
control variables), which should be reviewed first.
Consider the IV regression model in Equation (12.12) with a single X and a single W:
Yi = b0 + b1Xi + b2Wi + u i.
(12.22)
We replace IV regression assumption 1 in Key Concept 12.4 [which states that E1u i " Wi 2 = 04
with the assumption that, conditional on Wi, the mean of ui does not depend on Zi:
E1u i " Wi, Zi 2 = E1u i " Wi 2.
(12.23)
The next steps in the argument parallel those for regression with control variables in
Equations (6.23)–(6.25) in Appendix 6.5. Assume that E1u i " Wi 2 is linear in W i, so
E1u i "Wi 2 = g0 + g1Wi, where g0 and g1 are coefficients. Then
Yi = b0 + b1Xi + b2Wi + u i - E1u i " Wi, Zi 2 + E1u i " Wi, Zi 2
= b0 + b1Xi + b2Wi + ei + g0 + g1Wi
(12.24)
where the first line adds and subtracts E1u i " Wi, Zi 2 to the right hand side of Equation (12.22),
and the second line and defines ei = u i - E1u i " Wi, Zi 2 and uses the conditional mean inde-
pendence assumption plus linearity to write E1u i " Wi, Zi 2 = E1u i " Wi 2 = g0 + g1Wi. We thus
have that,
Yi = d0 + b1Xi + d1Wi + ei,
(12.25)
where d0 = b0 + g0 and d1 = b2 + g1. Now E1ei " Wi, Zi 2 = E3u i - E1u i " Wi, Zi 2 " Wi, Zi 4 =
E1u i " Wi, Zi 2 - E1u i " Wi, Zi 2 = 0, which in turn implies corr1Zi, ei 2 = 0. Thus IV regression
assumption 1 and the instrument exogeneity requirement (condition 2 in Key Concept 12.3)
both hold for Equation (12.24) with error term ei, Thus, if IV regression assumption 1 is replaced
by conditional mean independence in Equation (12.23), the original IV regression assumptions
in Key Concept 12.4 apply to the modified regression in Equation (12.25).
Because the IV regression assumptions of Key Concept 12.4 hold for Equation (12.25),
all the methods of inference (for both weak and strong instruments) discussed in this chapter
apply to Equation (12.25). In particular, if the instruments are strong, the coefficients in Equation (12.25) will be estimated consistently by TSLS and TSLS tests, and confidence intervals
will be valid.
Just as in OLS with control variables, in general the TSLS coefficient on the control variable W does not have a causal interpretation. TSLS consistently estimates d1 in Equation
(12.25), but d1 is the sum of b2, the direct causal effect of W, and g1, which reflects the correlation between W and the omitted factors in ui for which W controls.
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TSLS with Control Variables
431
In the cigarette consumption regressions in Table 12.1, it is tempting to interpret the coefficient on the 10-year change in log income as the income elasticity of demand. If, however,
income growth is correlated with increases in education and if more education reduces smoking, income growth would have its own causal effect (b2, the income elasticity) plus an effect
arising from its correlation with education 1g1 2. If the latter effect is negative 1g1 6 02, the
income coefficients in Table 12.1 (which estimate d1 = b2 + g1) would underestimate the
income elasticity. As long as the conditional mean independence assumption in Equation
(12.23) holds, however, the TSLS estimator of the price elasticity is consistent, even if the
estimate of the income elasticity is not.
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